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<div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px;width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%=, 101%, 101%, 0%);visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#00007D,#000000,#9999FF,#9999CC,#666699,#CCCCE6" masterhref=3D"master04.xml"> <p:shaperange href=3D"master04.xml#_x0000_s30721"/><![if !ppt]><p:shaperan=ge href=3D"master04.xml#_x0000_s30723"/> <div v:shape=3D"_x0000_s30723" class=3DO> <div style=3D'text-align:right;mso-char-wrap:1;mso-kinsoku-overflow:1; position:absolute;top:94.0%;left:72.65%;width:21.53%;height:3.5%'><span style=3D'font-family:"Arial Black";font-size:50%;mso-field-code:meta16'>2<=/span><span
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search tree</i></b></span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'> <=/span></span><span style=3D'position:absolute;top:28.73%;left:7.54%;width:77.73%;height:13.79=%'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>such that for every </span>=</span><span style=3D'position:absolute;top:43.1%;left:7.54%;width:83.77%;height:13.79%='><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>internal node v of T, </spa=n></span><span style=3D'position:absolute;top:57.47%;left:7.54%;width:77.73%;height:13.79=%'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>the </span><span lang=3DEN-=US style=3D'color:#3028FF;mso-ansi-language:EN-US'><i>heights of the </i></sp=an></span><span style=3D'position:absolute;top:71.83%;left:7.54%;width:92.45%;height:13.79=%'><span lang=3DEN-US style=3D'color:#3028FF;mso-ansi-language:EN-US'><i>children o=f v can differ </i></span></span><span style=3D'position:absolute;top:85.63%;left:=7.54%; width:77.73%;height:13.79%'><span lang=3DEN-US style=3D'color:#3028FF;mso-=ansi-language: EN-US'><i>by at most 1</i></span><span lang=3DEN-US style=3D'mso-ansi-lang=uage: EN-US'><i>.</i></span><span lang=3DEN-US style=3D'font-size:86%;mso-ansi-l=anguage: EN-US;display:none'><i> </i></span></span></div> <div style=3D'position:absolute;top:87.0%;left:4.3%;width:42.32%;height:5.=25%'><span style=3D'position:absolute;top:0%;left:8.84%;width:91.15%;height:100.0%'><=span style=3D'font-size:86%;visibility:hidden'><span class=3DHBB style=3D'posit=ion:absolute; left:-9.7%'>¨</span></span><span lang=3DEN-US style=3D'font-size:86%; mso-ansi-language:EN-US;mso-special-format:lastCR;display:none'> </spa=n></span></div> </div> <v:shapetype id=3D"_x0000_t202" coordsize=3D"21600,21600" o:spt=3D"202" pa=th=3D"m,l,21600r21600,l21600,xe"> <v:stroke joinstyle=3D"miter"/> <v:path gradientshapeok=3D"t" o:connecttype=3D"rect"/> </v:shapetype><v:shape id=3D"_x0000_s3079" type=3D"#_x0000_t202" style=3D'=position:absolute; left:354pt;top:6in;width:324pt;height:55.25pt' filled=3D"f" fillcolor=3D"=#99f [4]" stroked=3D"f" strokecolor=3D"black [1]"> <v:fill color2=3D"white [0]"/> <v:shadow color=3D"#00007d [2]"/> <v:textbox style=3D'mso-fit-shape-to-text:t'/> <p:animation number=3D"3" delay=3D"0" type=3D"noBuild" effect=3D"fly" dir=ection=3D"left" flag=3D"17408"/></v:shape><v:shape id=3D"_x0000_s3082" style=3D'position=:absolute; left:626.25pt;top:249.625pt;width:1.25pt;height:1.5pt' coordsize=3D"10,12" o:regroupid=3D"1" path=3D"m10,l5,,,6r5,6l10,xe" fillcolor=3D"black" strok=ed=3D"f"> <v:path arrowok=3D"t"/> </v:shape><![if !vml]><img border=3D0 v:shapes=3D"_x0000_s3082" src=3D"slide0002_image005.gif" style=3D'position:absolute;top:46.25%;left:=86.89%; width:.56%;height:.75%'><![endif]>
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<body lang=3DEN-US style=3D'margin:0px;background-color:black'onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()"onkeypress=3D"_KPH()" oa:clockstart=3D"onstart">
<div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px;width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%=, 101%, 101%, 0%);visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#00007D,#000000,#9999FF,#9999CC,#666699,#CCCCE6" masterhref=3D"master04.xml"> <p:shaperange href=3D"master04.xml#_x0000_s30721"/><![if !ppt]><p:shaperan=ge href=3D"master04.xml#_x0000_s30723"/> <div v:shape=3D"_x0000_s30723" class=3DO> <div style=3D'text-align:right;mso-char-wrap:1;mso-kinsoku-overflow:1; position:absolute;top:94.0%;left:72.65%;width:21.53%;height:3.5%'><span style=3D'font-family:"Arial Black";font-size:50%;mso-field-code:meta16'>3<=/span><span style=3D'font-family:"Arial Black";font-size:50%;mso-special-format:lastCR; display:none'> </span></div> </div> <p:shaperange href=3D"master04.xml#_x0000_s30724"/><![if !vml]><img border==3D0 v:shapes=3D"_x0000_s30724,_x0000_s30725,_x0000_s30726,_x0000_s30727,_x0000=_s30728,_x0000_s30729,_x0000_s30730,_x0000_s30731,_x0000_s30732,_x0000_s307=33" src=3D"master04_image001.gif" style=3D'position:absolute;top:0%;left:0%; width:100.37%;height:8.5%'><![endif]><p:shaperange href=3D"master04.xml#_x0000_s30736"/><![endif]><p:shaperange href=3D"master04.xml#_x0000_m30734"/><v:shape id=3D"_x0000_s4098" type=3D="#_x0000_m30734" style=3D'position:absolute;left:30pt;top:30pt;width:612pt;height:66pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape> <div v:shape=3D"_x0000_s4098" class=3DT style=3D'position:absolute;top:7.7=5%; left:5.24%;width:83.14%;height:8.5%'><span lang=3DEN-US style=3D'mso-ansi-=language: EN-US'>Height of an AVL Tree</span></div> <p:shaperange href=3D"master04.xml#_x0000_m30735"/><v:shape id=3D"_x0000_s=4099" type=3D"#_x0000_m30735" style=3D'position:absolute;left:30pt;top:126pt;wi=dth:666pt; height:372pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1" size=3D"half"/><p:animation n=umber=3D"2" delay=3D"0" type=3D"text" level=3D"1" effect=3D"dissolve"/><p:tag name==3D"___PPT2001" type=3D"binary" datasize=3D"36" dataoffset=3D"4"/></v:shape> <div v:shape=3D"_x0000_s4099" class=3DHB> <div id=3D"_x0000_s4099_0_69" style=3D'mso-line-spacing:"90 20 0";position=
:absolute; top:23.75%;left:5.24%;width:106.55%;height:11.5%;white-space:nowrap'><span style=3D'position:absolute;top:0%;left:3.51%;width:96.48%;height:52.17%'><=span style=3D'color:#3028FF'><span style=3D'mso-special-format:bullet;color:#99=99CC; mso-color-index:5;position:absolute;left:-3.64%;top:.3em;font-family:Wingd=ings; font-size:75%'>¨</span></span><span lang=3DEN-US style=3D'color:#3028F=F; mso-ansi-language:EN-US'><b>Proposition</b></span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>: The </span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'><b><i>height</i></b></span><span lang=3D=EN-US style=3D'mso-ansi-language:EN-US'> of an AVL tree T storing </span></span>=<span style=3D'position:absolute;top:50.0%;left:3.51%;width:81.54%;height:52.17%='><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>n keys is O(log n). </s=pan></span></div> <div id=3D"_x0000_s4099_69_198" style=3D'mso-line-spacing:"90 20 0";positi=on:absolute; top:36.25%;left:5.24%;width:102.43%;height:17.25%;white-space:nowrap'><span style=3D'position:absolute;top:0%;left:3.65%;width:96.34%;height:34.78%'><=span style=3D'color:#3028FF'><span style=3D'mso-special-format:bullet;color:#99=99CC; mso-color-index:5;position:absolute;left:-3.79%;top:.3em;font-family:Wingd=ings; font-size:75%'>¨</span></span><span lang=3DEN-US style=3D'color:#3028F=F; mso-ansi-language:EN-US'><b>Justification</b></span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>: The easiest way to approach this </spa=n></span><span style=3D'position:absolute;top:33.33%;left:3.65%;width:95.61%;height:34.78=%'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>problem is to find </span><=span lang=3DEN-US style=3D'mso-ansi-language:EN-US'><b>n(h):</b></span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'> the </span><span lang=3DEN=-US style=3D'mso-ansi-language:EN-US'><i>minimum number of </i></span></span><=span style=3D'position:absolute;top:65.21%;left:3.65%;width:84.82%;height:34.78=%'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'><i>nodes</i></span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'> in an AVL tree of height h=. </span></span></div> <div id=3D"_x0000_s4099_198_232" style=3D'mso-line-spacing:"90 20 0";posit=ion: absolute;top:54.25%;left:5.24%;width:90.63%;height:6.0%;white-space:nowrap='><span style=3D'position:absolute;top:0%;left:4.13%;width:95.86%;height:100.0%'><=span style=3D'mso-special-format:bullet;color:#9999CC;mso-color-index:5;positio=n: absolute;left:-4.31%;top:.3em;font-family:Wingdings;font-size:75%'>¨</=span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>We see that n(1) =3D 1 and =n(2) =3D 2 </span></span></div> <div id=3D"_x0000_s4099_232_369" style=3D'mso-line-spacing:"90 20 0";posit=ion: absolute;top:61.25%;left:5.24%;width:105.24%;height:17.25%;white-space:now=
rap'><span style=3D'position:absolute;top:0%;left:3.55%;width:90.92%;height:34.78%'><=span style=3D'mso-special-format:bullet;color:#9999CC;mso-color-index:5;positio=n: absolute;left:-3.91%;top:.3em;font-family:Wingdings;font-size:75%'>¨</=span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>For h ≥ 3, an AVL tre=e of height h contains the </span></span><span style=3D'position:absolute;top:3=1.88%; left:3.55%;width:96.44%;height:34.78%'><span lang=3DEN-US style=3D'mso-ans=i-language: EN-US'>root node, one AVL subtree of height h-1 and the </span></span><span style=3D'position:absolute;top:65.21%;left:3.55%;width:82.56%;height:34.78=%'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>other AVL subtree of height= h-1 or h-2. </span></span></div> <div id=3D"_x0000_s4099_369_401" style=3D'mso-line-spacing:"90 20 0";posit=ion: absolute;top:79.25%;left:5.24%;width:90.63%;height:6.0%;white-space:nowrap='><span style=3D'position:absolute;top:0%;left:4.13%;width:95.86%;height:100.0%'><=span style=3D'mso-special-format:bullet;color:#9999CC;mso-color-index:5;positio=n: absolute;left:-4.31%;top:.3em;font-family:Wingdings;font-size:75%'>¨</=span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>i.e. n(h) =3D 1 + n(h-1) + =n(h-2)</span></span></div> </div></p:slide></div>
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<div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px;width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%=, 101%, 101%, 0%);visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#00007D,#000000,#9999FF,#9999CC,#666699,#CCCCE6" masterhref=3D"master04.xml"> <p:shaperange href=3D"master04.xml#_x0000_s30721"/><![if !ppt]><p:shaperan=ge href=3D"master04.xml#_x0000_s30723"/> <div v:shape=3D"_x0000_s30723" class=3DO> <div style=3D'text-align:right;mso-char-wrap:1;mso-kinsoku-overflow:1; position:absolute;top:94.0%;left:72.65%;width:21.53%;height:3.5%'><span style=3D'font-family:"Arial Black";font-size:50%;mso-field-code:meta16'>4<=/span><span style=3D'font-family:"Arial Black";font-size:50%;mso-special-format:lastCR; display:none'> </span></div> </div> <p:shaperange href=3D"master04.xml#_x0000_s30724"/><![if !vml]><img border==3D0 v:shapes=3D"_x0000_s30724,_x0000_s30725,_x0000_s30726,_x0000_s30727,_x0000=_s30728,_x0000_s30729,_x0000_s30730,_x0000_s30731,_x0000_s30732,_x0000_s307=33" src=3D"master04_image001.gif" style=3D'position:absolute;top:0%;left:0%; width:100.37%;height:8.5%'><![endif]><p:shaperange href=3D"master04.xml#_x0000_s30736"/><![endif]><p:shaperange href=3D"master04.xml#_x0000_m30734"/><v:shape id=3D"_x0000_s26626" type==3D"#_x0000_m30734" style=3D'position:absolute;left:36pt;top:36pt;width:9in;height:66pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape> <div v:shape=3D"_x0000_s26626" class=3DT style=3D'position:absolute;top:8.=75%; left:5.99%;width:88.2%;height:8.5%'><span lang=3DEN-US style=3D'mso-ansi-l=anguage: EN-US'>Height of an AVL Tree (2)</span></div> <p:shaperange href=3D"master04.xml#_x0000_m30735"/><v:shape id=3D"_x0000_s=26627" type=3D"#_x0000_m30735" style=3D'position:absolute;left:42pt;top:114pt;wi=dth:624pt; height:390pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1" size=3D"half"/><p:animation n=umber=3D"2" delay=3D"0" type=3D"text" level=3D"1" effect=3D"dissolve"/></v:shape> <div v:shape=3D"_x0000_s26627"> <div id=3D"_x0000_s26627_0_34" class=3DHB style=3D'mso-line-spacing:"90 20= 0"; position:absolute;top:21.5%;left:6.92%;width:84.83%;height:6.0%;white-spac=e:
nowrap'><span style=3D'position:absolute;top:0%;left:4.41%;width:95.58%; height:100.0%'><span style=3D'mso-special-format:bullet;color:#9999CC; mso-color-index:5;position:absolute;left:-4.61%;top:.3em;font-family:Wingd=ings; font-size:75%'>¨</span><span lang=3DEN-US style=3D'mso-ansi-language:E=N-US'>Knowing n(h-1) >=3D n(h-2), we get </span></span></div> <div id=3D"_x0000_s26627_34_76" class=3DHB style=3D'mso-line-spacing:"90 2=0 0"; position:absolute;top:28.5%;left:6.92%;width:84.83%;height:6.0%;white-spac=e: nowrap'><span style=3D'mso-special-format:nobullet;display:none;color:#999=9CC; mso-color-index:5;font-family:Wingdings;font-size:75%'>¨</span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'><span style=3D'mso-spacerun:yes'> </span>n(h) =3D n(h-1)= + n(h-2) + 1 > 2n(h-2) </span></div> <div id=3D"_x0000_s26627_76_91" class=3DHB1 style=3D'mso-line-spacing:"90 =20 0"; position:absolute;top:35.25%;left:11.79%;width:79.77%;height:6.0%;white-sp=ace: nowrap'><span style=3D'mso-special-format:nobullet;display:none;color:#999=9CC; mso-color-index:5;font-family:Wingdings;font-size:75%'>¨</span><span lang=3DEN-US style=3D'font-size:117%;color:#3028FF;mso-ansi-language:EN-US='>n(h) > 2n(h-2) </span></div> <div id=3D"_x0000_s26627_91_108" class=3DHB1 style=3D'mso-line-spacing:"90= 20 0"; position:absolute;top:42.25%;left:11.79%;width:79.77%;height:6.0%;white-sp=ace: nowrap'><span style=3D'mso-special-format:nobullet;display:none;color:#999=9CC; mso-color-index:5;font-family:Wingdings;font-size:75%'>¨</span><span lang=3DEN-US style=3D'font-size:117%;color:#3028FF;mso-ansi-language:EN-US='><span style=3D'mso-spacerun:yes'> </span>> 4n(h-4) </span></div> <div id=3D"_x0000_s26627_108_125" class=3DHB1 style=3D'mso-line-spacing:"9=0 20 0"; position:absolute;top:49.0%;left:11.79%;width:79.77%;height:6.0%;white-spa=ce: nowrap'><span style=3D'mso-special-format:nobullet;display:none;color:#999=9CC; mso-color-index:5;font-family:Wingdings;font-size:75%'>¨</span><span lang=3DEN-US style=3D'font-size:117%;color:#3028FF;mso-ansi-language:EN-US='><span style=3D'mso-spacerun:yes'> </span>> 8n(h-6) </span></div> <div id=3D"_x0000_s26627_125_134" class=3DHB1 style=3D'mso-line-spacing:"9=0 20 0"; position:absolute;top:56.0%;left:11.79%;width:79.77%;height:6.0%;white-spa=ce: nowrap'><span style=3D'mso-special-format:nobullet;display:none;color:#999=9CC; mso-color-index:5;font-family:Wingdings;font-size:75%'>¨</span><span lang=3DEN-US style=3D'font-size:117%;color:#3028FF;mso-ansi-language:EN-US='><span style=3D'mso-spacerun:yes'> </span>… </span></div> <div id=3D"_x0000_s26627_134_153" class=3DHB1 style=3D'mso-line-spacing:"9=0 20 0"; position:absolute;top:62.75%;left:11.79%;width:79.77%;height:6.0%;white-sp=
ace: nowrap'><span style=3D'mso-special-format:nobullet;display:none;color:#999=9CC; mso-color-index:5;font-family:Wingdings;font-size:75%'>¨</span><span lang=3DEN-US style=3D'font-size:117%;color:#3028FF;mso-ansi-language:EN-US='><span style=3D'mso-spacerun:yes'> </span>>=; 2</span><span lang=3DEN-US style=3D'font-size:78%;color:#3028FF;position:relative;top:-.=45em; mso-text-raise:30%;mso-ansi-language:EN-US'>i</span><span lang=3DEN-US style=3D'font-size:117%;color:#3028FF;mso-ansi-language:EN-US'>n(h-2i)</sp=an><span lang=3DEN-US style=3D'font-size:117%;mso-ansi-language:EN-US;display:none'=> </span></div> <div id=3D"_x0000_s26627_153_201" class=3DHB style=3D'mso-line-spacing:"90= 20 0"; position:absolute;top:69.5%;left:6.92%;width:96.25%;height:6.0%;white-spac=e: nowrap'><span style=3D'position:absolute;top:0%;left:3.89%;width:96.1%; height:100.0%'><span style=3D'mso-special-format:bullet;color:#9999CC; mso-color-index:5;position:absolute;left:-4.04%;top:.3em;font-family:Wingd=ings; font-size:75%'>¨</span><span lang=3DEN-US style=3D'mso-ansi-language:E=N-US'>When i =3D h/2-1 we get: n(h) > 2</span><span lang=3DEN-US style=3D'font-siz=e:67%; position:relative;top:-.45em;mso-text-raise:30%;mso-ansi-language:EN-US'>h=/2-1</span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>n(2) =3D 2</span><span lang==3DEN-US style=3D'font-size:67%;position:relative;top:-.45em;mso-text-raise:30%; mso-ansi-language:EN-US'>h/2 </span></span></div> <div id=3D"_x0000_s26627_201_234" class=3DHB style=3D'mso-line-spacing:"90= 20 0"; position:absolute;top:76.5%;left:6.92%;width:84.83%;height:6.0%;white-spac=e: nowrap'><span style=3D'position:absolute;top:0%;left:4.41%;width:95.58%; height:100.0%'><span style=3D'mso-special-format:bullet;color:#9999CC; mso-color-index:5;position:absolute;left:-4.61%;top:.3em;font-family:Wingd=ings; font-size:75%'>¨</span><span lang=3DEN-US style=3D'mso-ansi-language:E=N-US'>Taking logarithms: h < 2log n(h) </span></span></div> <div id=3D"_x0000_s26627_234_277" class=3DHB style=3D'mso-line-spacing:"90= 20 0"; position:absolute;top:83.25%;left:6.92%;width:90.26%;height:6.0%;white-spa=ce: nowrap'><span style=3D'position:absolute;top:0%;left:4.14%;width:95.85%; height:100.0%'><span style=3D'mso-special-format:bullet;color:#9999CC; mso-color-index:5;position:absolute;left:-4.32%;top:.3em;font-family:Wingd=ings; font-size:75%'>¨</span><span lang=3DEN-US style=3D'mso-ansi-language:E=N-US'>Thus the height of an AVL tree is O(log n)</span></span></div> </div></p:slide></div>
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<head><meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"><meta name=3DProgId content=3DPowerPoint.Slide><meta name=3DGenerator content=3D"Microsoft PowerPoint 10"><link id=3DMain-File rel=3DMain-File href=3D"../lec11.htm"><link rel=3DPreview href=3Dpreview.wmf><!--[if !mso]><style>v\:* {behavior:url(#default#VML);}o\:* {behavior:url(#default#VML);}p\:* {behavior:url(#default#VML);}.shape {behavior:url(#default#VML);}v\:textbox {display:none;}</style><![endif]--><title>AVL Trees</title><![if IE]><base href=3D"file:///C:\D0F41251\lec11_files\slide0027.htm"id=3D"webarch_temp_base_tag"><![endif]><meta name=3DDescription content=3D"2/17/2005: A sharper bound"><link rel=3DStylesheet href=3D"master04_stylesheet.css"><!--[if gte IE 5]><object id=3DMSOANIM classid=3D"CLSID:A4639D2F-774E-11D3-A490-00C04F6843FB"=></object><object id=3DMSOTIME classid=3D"CLSID:A4639D29-774E-11D3-A490-00C04F6843FB"=
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<div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px;width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%=, 101%, 101%, 0%);visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#00007D,#000000,#9999FF,#9999CC,#666699,#CCCCE6" masterhref=3D"master04.xml"> <p:shaperange href=3D"master04.xml#_x0000_s30721"/><![if !ppt]><p:shaperan=ge href=3D"master04.xml#_x0000_s30723"/> <div v:shape=3D"_x0000_s30723" class=3DO> <div style=3D'text-align:right;mso-char-wrap:1;mso-kinsoku-overflow:1; position:absolute;top:94.0%;left:72.65%;width:21.53%;height:3.5%'><span style=3D'font-family:"Arial Black";font-size:50%;mso-field-code:meta16'>5<=/span><span style=3D'font-family:"Arial Black";font-size:50%;mso-special-format:lastCR; display:none'> </span></div> </div> <p:shaperange href=3D"master04.xml#_x0000_s30724"/><![if !vml]><img border==3D0 v:shapes=3D"_x0000_s30724,_x0000_s30725,_x0000_s30726,_x0000_s30727,_x0000=_s30728,_x0000_s30729,_x0000_s30730,_x0000_s30731,_x0000_s30732,_x0000_s307=33" src=3D"master04_image001.gif" style=3D'position:absolute;top:0%;left:0%; width:100.37%;height:8.5%'><![endif]><p:shaperange href=3D"master04.xml#_x0000_s30736"/><![endif]><p:shaperange href=3D"master04.xml#_x0000_m30734"/><v:shape id=3D"_x0000_s33794" type==3D"#_x0000_m30734" style=3D'position:absolute;left:36pt;top:36pt;width:9in;height:66pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/>
<v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape> <div v:shape=3D"_x0000_s33794" class=3DT style=3D'position:absolute;top:8.=75%; left:5.99%;width:88.2%;height:8.5%'>A sharper bound</div> <p:shaperange href=3D"master04.xml#_x0000_m30735"/><v:shape id=3D"_x0000_s=33795" type=3D"#_x0000_m30735" style=3D'position:absolute;left:36pt;top:120pt;wi=dth:660pt; height:384pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/><p:animation number=3D"2" de=lay=3D"0" type=3D"object" effect=3D"cut" direction=3D"noBlack" flag=3D"17408"/></v=:shape> <div v:shape=3D"_x0000_s33795" class=3DB style=3D'position:absolute;top:23=.25%; left:5.99%;width:106.92%;height:70.75%'> <div id=3D"_x0000_s33795_0_73" style=3D'position:absolute;top:0%;left:0%; width:85.98%;height:19.78%;white-space:nowrap'><span style=3D'position:abs=olute; top:0%;left:4.07%;width:95.92%;height:48.21%'><span class=3DBB style=3D'po=sition: absolute;left:-4.24%;top:.3em'>¨</span>We will show how to obtain a sharper </span><span style=3D'position:absolute;top:51.78%;left:4.07%; width:93.68%;height:48.21%'>bound on the height of an AVL tree. </span=></div> <div id=3D"_x0000_s33795_73_199" style=3D'position:absolute;top:22.26%;lef=t:0%; width:100.0%;height:29.68%;white-space:nowrap'><span style=3D'position:abs=olute; top:0%;left:3.5%;width:95.62%;height:32.14%'><span class=3DBB style=3D'pos=ition: absolute;left:-3.66%;top:.3em'>¨</span>We prove using induction that t=he minimum </span><span style=3D'position:absolute;top:33.33%;left:3.5%;width=:96.49%; height:32.14%'>number of nodes in an AVL tree of height h, </span><span style=3D'position:absolute;top:67.85%;left:3.5%;width:88.79%;height:32.14%='>n(h) >=3D c<span style=3D'mso-hansi-font-family:Arial;font-size:66%;position=:relative; top:-.45em;mso-text-raise:30%'>h</span>,<span style=3D'mso-spacerun:yes'>&=nbsp; </span>where c is some number >1. </span></div> <div id=3D"_x0000_s33795_199_234" style=3D'position:absolute;top:54.41%; left:0%;width:84.06%;height:9.54%;white-space:nowrap'><span style=3D'posit=ion: absolute;top:0%;left:4.16%;width:95.83%;height:100.0%'><span class=3DBB style=3D'position:absolute;left:-4.34%;top:.3em'>¨</span>Base case: h==3D1. Now n(h) >=3D c > 1. </span></div> <div id=3D"_x0000_s33795_234_270" style=3D'position:absolute;top:66.43%; left:0%;width:84.06%;height:9.54%;white-space:nowrap'><span style=3D'posit=ion: absolute;top:0%;left:4.16%;width:95.83%;height:100.0%'><span class=3DBB style=3D'position:absolute;left:-4.34%;top:.3em'>¨</span>Suppose claim= is true for all h < k </span></div> <div id=3D"_x0000_s33795_270_302" style=3D'position:absolute;top:78.44%; left:0%;width:84.06%;height:9.54%;white-space:nowrap'><span style=3D'posit=
ion: absolute;top:0%;left:4.16%;width:95.83%;height:100.0%'><span class=3DBB style=3D'position:absolute;left:-4.34%;top:.3em'>¨</span>We have to sh=ow that n(k) >=3D c<span style=3D'mso-hansi-font-family:Arial;font-size:66=%; position:relative;top:-.45em;mso-text-raise:30%'>k</span> </span></div> <div style=3D'position:absolute;top:90.45%;left:0%;width:84.06%;height:9.5=4%'><span style=3D'position:absolute;top:0%;left:4.16%;width:95.83%;height:100.0%'><=span style=3D'visibility:hidden'><span class=3DBB style=3D'position:absolute;le=ft:-4.34%'>¨</span></span><span style=3D'mso-special-format:lastCR;display:none'> </span></span></div> </div></p:slide></div>
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<html xmlns:v=3D"urn:schemas-microsoft-com:vml"xmlns:o=3D"urn:schemas-microsoft-com:office:office"xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint"xmlns:oa=3D"urn:schemas-microsoft-com:office:activation"xmlns=3D"http://www.w3.org/TR/REC-html40">
<head><meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"><meta name=3DProgId content=3DPowerPoint.Slide><meta name=3DGenerator content=3D"Microsoft PowerPoint 10"><link id=3DMain-File rel=3DMain-File href=3D"../lec11.htm"><link rel=3DPreview href=3Dpreview.wmf><!--[if !mso]><style>v\:* {behavior:url(#default#VML);}o\:* {behavior:url(#default#VML);}p\:* {behavior:url(#default#VML);}.shape {behavior:url(#default#VML);}v\:textbox {display:none;}</style><![endif]--><title>AVL Trees</title><![if IE]><base href=3D"file:///C:\D0F41251\lec11_files\slide0028.htm"id=3D"webarch_temp_base_tag"><![endif]><meta name=3DDescription content=3D"2/17/2005: Sharper bound (2)"><link rel=3DStylesheet href=3D"master04_stylesheet.css"><!--[if gte IE 5]><object id=3DMSOANIM classid=3D"CLSID:A4639D2F-774E-11D3-A490-00C04F6843FB"=></object>
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<body lang=3DEN-US style=3D'margin:0px;background-color:black'onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()"onkeypress=3D"_KPH()" oa:clockstart=3D"onstart">
<div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px;width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%=, 101%, 101%, 0%);visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#00007D,#000000,#9999FF,#9999CC,#666699,#CCCCE6" masterhref=3D"master04.xml"> <p:shaperange href=3D"master04.xml#_x0000_s30721"/><![if !ppt]><p:shaperan=ge href=3D"master04.xml#_x0000_s30723"/> <div v:shape=3D"_x0000_s30723" class=3DO> <div style=3D'text-align:right;mso-char-wrap:1;mso-kinsoku-overflow:1; position:absolute;top:94.0%;left:72.65%;width:21.53%;height:3.5%'><span style=3D'font-family:"Arial Black";font-size:50%;mso-field-code:meta16'>6<=/span><span style=3D'font-family:"Arial Black";font-size:50%;mso-special-format:lastCR; display:none'> </span></div> </div> <p:shaperange href=3D"master04.xml#_x0000_s30724"/><![if !vml]><img border==3D0 v:shapes=3D"_x0000_s30724,_x0000_s30725,_x0000_s30726,_x0000_s30727,_x0000=_s30728,_x0000_s30729,_x0000_s30730,_x0000_s30731,_x0000_s30732,_x0000_s307=33" src=3D"master04_image001.gif" style=3D'position:absolute;top:0%;left:0%; width:100.37%;height:8.5%'><![endif]><p:shaperange href=3D"master04.xml#_x0000_s30736"/><![endif]><p:shaperange href=3D"master04.xml#_x0000_m30734"/><v:shape id=3D"_x0000_s34818" type==3D"#_x0000_m30734"
style=3D'position:absolute;left:36pt;top:36pt;width:9in;height:66pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape> <div v:shape=3D"_x0000_s34818" class=3DT style=3D'position:absolute;top:8.=75%; left:5.99%;width:88.2%;height:8.5%'>Sharper bound (2)</div> <p:shaperange href=3D"master04.xml#_x0000_m30735"/><v:shape id=3D"_x0000_s=34819" type=3D"#_x0000_m30735" style=3D'position:absolute;left:36pt;top:102pt;wi=dth:654pt; height:402pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1" size=3D"half"/><p:animation n=umber=3D"2" delay=3D"0" type=3D"text" level=3D"1" effect=3D"cut" direction=3D"noBlac=k"/></v:shape> <div v:shape=3D"_x0000_s34819" class=3DHB> <div id=3D"_x0000_s34819_0_23" style=3D'mso-line-spacing:"90 20 0";positio=n:absolute; top:19.5%;left:5.99%;width:88.95%;height:6.0%;white-space:nowrap'><span style=3D'position:absolute;top:0%;left:4.21%;width:95.78%;height:100.0%'><=span style=3D'mso-special-format:bullet;color:#9999CC;mso-color-index:5;positio=n: absolute;left:-4.39%;top:.3em;font-family:Wingdings;font-size:75%'>¨</=span>n(k) =3D n(k-1)+n(k-2)+1 </span></div> <div id=3D"_x0000_s34819_23_76" style=3D'mso-line-spacing:"90 20 0";positi=on:absolute; top:26.25%;left:5.99%;width:94.19%;height:6.0%;white-space:nowrap'><span style=3D'mso-special-format:nobullet;display:none;color:#9999CC;mso-color-=index: 5;font-family:Wingdings;font-size:75%'>¨</span><span style=3D'mso-spacerun:yes'>  =; </span>>=3D c<span style=3D'mso-hansi-font-family:Arial;font-size:67%; position:relative;top:-.45em;mso-text-raise:30%'>k-1</span><span style=3D'mso-spacerun:yes'> </span>+ c<span style=3D'mso-hansi-font-f=amily: Arial;font-size:67%;position:relative;top:-.45em;mso-text-raise:30%'>k-2</=span><span style=3D'mso-spacerun:yes'> </span>(by induction hypothesis) </d=iv> <div id=3D"_x0000_s34819_76_155" style=3D'mso-line-spacing:"90 20 0";posit=ion: absolute;top:33.0%;left:5.99%;width:102.99%;height:11.5%;white-space:nowra=p'><span style=3D'position:absolute;top:0%;left:3.63%;width:96.36%;height:52.17%'><=span style=3D'mso-special-format:bullet;color:#9999CC;mso-color-index:5;positio=n: absolute;left:-3.77%;top:.3em;font-family:Wingdings;font-size:75%'>¨</=span>We will be able to show that n(k) >=3D c<span style=3D'mso-hansi-font-fami=ly:Arial; font-size:67%;position:relative;top:-.45em;mso-text-raise:30%'>k</span><sp=an style=3D'mso-spacerun:yes'> </span>if we can </span><span style=3D'po=sition:
absolute;top:50.0%;left:3.63%;width:82.72%;height:52.17%'>show that c<span style=3D'mso-hansi-font-family:Arial;font-size:67%;position:relative;top:-=.45em; mso-text-raise:30%'>k-1</span><span style=3D'mso-spacerun:yes'> </spa=n>+ c<span style=3D'mso-hansi-font-family:Arial;font-size:67%;position:relative;top:-=.45em; mso-text-raise:30%'>k-2</span><span style=3D'mso-spacerun:yes'> </span>>=3D c<span style=3D'mso-hansi-=font-family: Arial;font-size:67%;position:relative;top:-.45em;mso-text-raise:30%'>k</sp=an>. </span></div> <div id=3D"_x0000_s34819_155_193" style=3D'mso-line-spacing:"90 20 0";posi=tion: absolute;top:45.5%;left:5.99%;width:88.95%;height:6.0%;white-space:nowrap'=><span style=3D'position:absolute;top:0%;left:4.21%;width:95.78%;height:100.0%'><=span style=3D'mso-special-format:bullet;color:#9999CC;mso-color-index:5;positio=n: absolute;left:-4.39%;top:.3em;font-family:Wingdings;font-size:75%'>¨</=span>So c should be such that c<span style=3D'mso-hansi-font-family:Arial;font-siz=e: 67%;position:relative;top:-.45em;mso-text-raise:30%'>2</span>-c-1 <=3D 0. </span></div> <div id=3D"_x0000_s34819_193_230" style=3D'mso-line-spacing:"90 20 0";posi=tion: absolute;top:52.5%;left:5.99%;width:88.95%;height:6.0%;white-space:nowrap'=><span style=3D'position:absolute;top:0%;left:4.21%;width:95.78%;height:100.0%'><=span style=3D'mso-special-format:bullet;color:#9999CC;mso-color-index:5;positio=n: absolute;left:-4.39%;top:.3em;font-family:Wingdings;font-size:75%'>¨</=span>The quadratic equation c<span style=3D'mso-hansi-font-family:Arial;font-size:6=7%; position:relative;top:-.45em;mso-text-raise:30%'>2</span>-c-1=3D0 has 
=;</span></div> <div id=3D"_x0000_s34819_230_268" style=3D'mso-line-spacing:"90 20 0";posi=tion: absolute;top:59.25%;left:5.99%;width:88.95%;height:6.0%;white-space:nowrap='><span style=3D'mso-special-format:nobullet;display:none;color:#9999CC;mso-color-=index: 5;font-family:Wingdings;font-size:75%'>¨</span><span style=3D'mso-spacerun:yes'> </span>roots<span style=3D'mso-spacerun:yes'>  =; </span>and<span style=3D'mso-spacerun:yes'>  =; </span>. </div> <div id=3D"_x0000_s34819_268_324" style=3D'mso-line-spacing:"90 20 0";posi=tion: absolute;top:66.0%;left:5.99%;width:102.24%;height:11.5%;white-space:nowra=p'><span style=3D'position:absolute;top:0%;left:3.66%;width:96.33%;height:52.17%'><=span style=3D'mso-special-format:bullet;color:#9999CC;mso-color-index:5;positio=n: absolute;left:-3.8%;top:.3em;font-family:Wingdings;font-size:75%'>¨</s=pan>Hence
we can take c as<span style=3D'mso-spacerun:yes'>  =; </span>which is roughly </span><span style=3D'position:absolute;top:50.0%; left:3.66%;width:83.33%;height:52.17%'>1.63 </span></div> <div id=3D"_x0000_s34819_324_378" style=3D'mso-line-spacing:"90 20 0";posi=tion: absolute;top:78.5%;left:5.99%;width:99.62%;height:11.5%;white-space:nowrap='><span style=3D'position:absolute;top:0%;left:3.75%;width:96.24%;height:52.17%'><=span style=3D'mso-special-format:bullet;color:#9999CC;mso-color-index:5;positio=n: absolute;left:-3.9%;top:.3em;font-family:Wingdings;font-size:75%'>¨</s=pan>Hence AVL tree on n nodes has height atmost </span><span style=3D'position:absol=ute; top:50.0%;left:3.75%;width:85.52%;height:52.17%'>log<span style=3D'mso-han=si-font-family: Arial;font-size:67%;position:relative;top:.37em;mso-text-raise:-25%'>1.63<=/span><span style=3D'mso-spacerun:yes'> </span>n<span style=3D'mso-hansi-font-fam=ily: Arial;font-size:67%;position:relative;top:-.45em;mso-text-raise:30%; mso-special-format:lastCR;display:none'> </span></span></div> </div> <v:shapetype id=3D"_x0000_t75" coordsize=3D"21600,21600" o:spt=3D"75" o:preferrelative=3D"t" path=3D"m@4@5l@4@11@9@11@9@5xe" filled=3D"f" strok=ed=3D"f"> <v:stroke joinstyle=3D"miter"/> <v:formulas> <v:f eqn=3D"if lineDrawn pixelLineWidth 0"/> <v:f eqn=3D"sum @0 1 0"/> <v:f eqn=3D"sum 0 0 @1"/> <v:f eqn=3D"prod @2 1 2"/> <v:f eqn=3D"prod @3 21600 pixelWidth"/> <v:f eqn=3D"prod @3 21600 pixelHeight"/> <v:f eqn=3D"sum @0 0 1"/> <v:f eqn=3D"prod @6 1 2"/> <v:f eqn=3D"prod @7 21600 pixelWidth"/> <v:f eqn=3D"sum @8 21600 0"/> <v:f eqn=3D"prod @7 21600 pixelHeight"/> <v:f eqn=3D"sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok=3D"f" gradientshapeok=3D"t" o:connecttype=3D"rect"/> <o:lock v:ext=3D"edit" aspectratio=3D"t"/> </v:shapetype><v:shape id=3D"_x0000_s34821" type=3D"#_x0000_t75" style=3D'=position:absolute; left:2in;top:312pt;width:78pt;height:48.375pt;visibility:visible; mso-wrap-style:none;v-text-anchor:middle' fillcolor=3D"#99f [4]" strokeco=lor=3D"black [1]"> <v:fill color2=3D"white [0]"/> <v:imagedata src=3D"slide0028_image013.png" o:title=3D"txp_fig"/> <v:shadow color=3D"#00007d [2]"/> <p:animation number=3D"8" delay=3D"0" type=3D"object" effect=3D"cut" direction=3D"noBlack" flag=3D"16388"/><p:tag name=3D"SOURCE" type=3D"str=ing" value=3D"\documentclass{slides}\pagestyle{empty} \begin{documen=t} $\frac{1-\sqrt{5}}{2}$ \end{document} "/><p:t=ag name=3D"EXTERNALNAME" type=3D"string" value=3D"txp_fig"/><p:tag name=3D"=BLEND" type=3D"string" value=3D"False"/><p:tag name=3D"TRANSPARENT" type=3D"str=ing"
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</div> <p:shaperange href=3D"master04.xml#_x0000_s30724"/><![if !vml]><img border==3D0 v:shapes=3D"_x0000_s30724,_x0000_s30725,_x0000_s30726,_x0000_s30727,_x0000=_s30728,_x0000_s30729,_x0000_s30730,_x0000_s30731,_x0000_s30732,_x0000_s307=33" src=3D"master04_image001.gif" style=3D'position:absolute;top:0%;left:0%; width:100.37%;height:8.5%'><![endif]><p:shaperange href=3D"master04.xml#_x0000_s30736"/><![endif]><p:shaperange href=3D"master04.xml#_x0000_m30734"/><v:shape id=3D"_x0000_s36866" type==3D"#_x0000_m30734" style=3D'position:absolute;left:36pt;top:36pt;width:9in;height:60pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape> <div v:shape=3D"_x0000_s36866" class=3DT style=3D'position:absolute;top:8.=25%; left:5.99%;width:88.2%;height:8.5%'>Structure of an AVL tree</div> <p:shaperange href=3D"master04.xml#_x0000_m30735"/><v:shape id=3D"_x0000_s=36867" type=3D"#_x0000_m30735" style=3D'position:absolute;left:36pt;top:120pt;wi=dth:9in; height:384pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/><p:animation number=3D"2" de=lay=3D"0" type=3D"text" level=3D"1" effect=3D"cut" direction=3D"noBlack"/></v:shap=e> <div v:shape=3D"_x0000_s36867" class=3DB> <div id=3D"_x0000_s36867_0_33" style=3D'position:absolute;top:23.25%;left:=5.99%; width:88.2%;height:6.75%;white-space:nowrap'><span style=3D'position:absol=ute; top:0%;left:4.24%;width:95.75%;height:100.0%'><span class=3DBB style=3D'po=sition: absolute;left:-4.43%;top:.3em'>¨</span>Consider an AVL tree on n nodes. </span></div> <div id=3D"_x0000_s36867_33_79" style=3D'position:absolute;top:31.75%;left=:5.99%; width:93.07%;height:14.0%;white-space:nowrap'><span style=3D'position:abso=lute; top:0%;left:4.02%;width:95.97%;height:48.21%'><span class=3DBB style=3D'po=sition: absolute;left:-4.19%;top:.3em'>¨</span>Consider a leaf which is closes=t to the </span><span style=3D'position:absolute;top:51.78%;left:4.02%;width:90=.74%; height:48.21%'>root. </span></div> <div id=3D"_x0000_s36867_79_112" style=3D'position:absolute;top:47.5%;left=:5.99%; width:88.2%;height:6.75%;white-space:nowrap'><span style=3D'position:absol=ute; top:0%;left:4.24%;width:95.75%;height:100.0%'><span class=3DBB style=3D'po=sition: absolute;left:-4.43%;top:.3em'>¨</span>Suppose this leaf is at level k. </span></div> <div id=3D"_x0000_s36867_112_170" style=3D'position:absolute;top:56.0%;lef=t:5.99%; width:100.74%;height:14.0%;white-space:nowrap'><span style=3D'position:abs=olute;
top:0%;left:3.71%;width:96.28%;height:48.21%'><span class=3DBB style=3D'po=sition: absolute;left:-3.86%;top:.3em'>¨</span>We will show that the height of= the tree is </span><span style=3D'position:absolute;top:50.0%;left:3.71%;width=:83.82%; height:48.21%'>at most 2k-1. </span></div> <div style=3D'position:absolute;top:71.5%;left:5.99%;width:88.2%;height:6.=75%'><span style=3D'position:absolute;top:0%;left:4.24%;width:95.75%;height:100.0%'><=span style=3D'visibility:hidden'><span class=3DBB style=3D'position:absolute;le=ft:-4.43%'>¨</span></span><span style=3D'mso-special-format:lastCR;display:none'> </span></span></div> </div></p:slide></div>
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<head><meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"><meta name=3DProgId content=3DPowerPoint.Slide><meta name=3DGenerator content=3D"Microsoft PowerPoint 10"><link id=3DMain-File rel=3DMain-File href=3D"../lec11.htm"><link rel=3DPreview href=3Dpreview.wmf><!--[if !mso]><style>v\:* {behavior:url(#default#VML);}o\:* {behavior:url(#default#VML);}p\:* {behavior:url(#default#VML);}.shape {behavior:url(#default#VML);}v\:textbox {display:none;}</style><![endif]--><title>AVL Trees</title><![if IE]><base href=3D"file:///C:\D0F41251\lec11_files\slide0030.htm"id=3D"webarch_temp_base_tag"><![endif]><meta name=3DDescription content=3D"2/17/2005: Structure of an AVL tree (2)="><link rel=3DStylesheet href=3D"master04_stylesheet.css"><!--[if gte IE 5]><object id=3DMSOANIM classid=3D"CLSID:A4639D2F-774E-11D3-A490-00C04F6843FB"=></object><object id=3DMSOTIME classid=3D"CLSID:A4639D29-774E-11D3-A490-00C04F6843FB"=></object><![endif]--><![if !ppt]><style media=3Dprint><!--.sld
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<div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px;width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%=, 101%, 101%, 0%);visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#00007D,#000000,#9999FF,#9999CC,#666699,#CCCCE6" masterhref=3D"master04.xml"> <p:shaperange href=3D"master04.xml#_x0000_s30721"/><![if !ppt]><p:shaperan=ge href=3D"master04.xml#_x0000_s30723"/> <div v:shape=3D"_x0000_s30723" class=3DO> <div style=3D'text-align:right;mso-char-wrap:1;mso-kinsoku-overflow:1; position:absolute;top:94.0%;left:72.65%;width:21.53%;height:3.5%'><span style=3D'font-family:"Arial Black";font-size:50%;mso-field-code:meta16'>8<=/span><span style=3D'font-family:"Arial Black";font-size:50%;mso-special-format:lastCR; display:none'> </span></div> </div> <p:shaperange href=3D"master04.xml#_x0000_s30724"/><![if !vml]><img border==3D0 v:shapes=3D"_x0000_s30724,_x0000_s30725,_x0000_s30726,_x0000_s30727,_x0000=_s30728,_x0000_s30729,_x0000_s30730,_x0000_s30731,_x0000_s30732,_x0000_s307=33" src=3D"master04_image001.gif" style=3D'position:absolute;top:0%;left:0%; width:100.37%;height:8.5%'><![endif]><p:shaperange href=3D"master04.xml#_x0000_s30736"/><![endif]><p:shaperange href=3D"master04.xml#_x0000_m30734"/><v:shape id=3D"_x0000_s37890" type==3D"#_x0000_m30734" style=3D'position:absolute;left:36pt;top:36pt;width:9in;height:60pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape> <div v:shape=3D"_x0000_s37890" class=3DT style=3D'position:absolute;top:8.=
25%; left:5.99%;width:88.2%;height:8.5%'>Structure of an AVL tree (2)</div> <p:shaperange href=3D"master04.xml#_x0000_m30735"/><v:shape id=3D"_x0000_s=37891" type=3D"#_x0000_m30735" style=3D'position:absolute;left:30pt;top:102pt;wi=dth:9in; height:384pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/><p:animation number=3D"2" de=lay=3D"0" type=3D"text" level=3D"1" effect=3D"cut" direction=3D"noBlack"/></v:shap=e> <div v:shape=3D"_x0000_s37891" class=3DB> <div id=3D"_x0000_s37891_0_84" style=3D'mso-line-spacing:"90 20 0";positio=n:absolute; top:19.5%;left:5.24%;width:104.49%;height:11.5%;white-space:nowrap'><span style=3D'position:absolute;top:0%;left:3.58%;width:96.41%;height:52.17%'><=span style=3D'font-size:88%'><span style=3D'mso-special-format:bullet;color:#99=99CC; mso-color-index:5;position:absolute;left:-3.71%;top:.3em;font-family:Wingd=ings; font-size:75%'>¨</span></span><span style=3D'font-size:88%'>Claim: Sin=ce closest leaf is at level k all nodes at </span></span><span style=3D'posit=ion: absolute;top:47.82%;left:3.58%;width:80.82%;height:52.17%'><span style=3D'font-size:88%'>levels 1..k-2 have 2 children. </span></span><=/div> <div id=3D"_x0000_s37891_84_110" style=3D'mso-line-spacing:"90 20 0";posit=ion: absolute;top:31.75%;left:5.24%;width:88.2%;height:6.0%;white-space:nowrap'=><span style=3D'position:absolute;top:0%;left:4.24%;width:95.75%;height:100.0%'><=span style=3D'font-size:88%'><span style=3D'mso-special-format:bullet;color:#99=99CC; mso-color-index:5;position:absolute;left:-4.43%;top:.3em;font-family:Wingd=ings; font-size:75%'>¨</span></span><span style=3D'font-size:88%'>Proof is by contradiction </span></span></div> <div id=3D"_x0000_s37891_110_159" style=3D'mso-line-spacing:"90 20 0";posi=tion: absolute;top:38.75%;left:5.24%;width:101.68%;height:6.0%;white-space:nowra=p'><span style=3D'position:absolute;top:0%;left:3.68%;width:96.31%;height:100.0%'><=span style=3D'font-size:88%'><span style=3D'mso-special-format:bullet;color:#99=99CC; mso-color-index:5;position:absolute;left:-3.82%;top:.3em;font-family:Wingd=ings; font-size:75%'>¨</span></span><span style=3D'font-size:88%'>Suppose no=de u at level k-2 has only 1 child, v. </span></span></div> <div id=3D"_x0000_s37891_159_202" style=3D'mso-line-spacing:"90 20 0";posi=tion: absolute;top:45.5%;left:5.24%;width:88.2%;height:6.0%;white-space:nowrap'>=<span style=3D'position:absolute;top:0%;left:4.24%;width:95.75%;height:100.0%'><=span style=3D'font-size:88%'><span style=3D'mso-special-format:bullet;color:#99=99CC;
mso-color-index:5;position:absolute;left:-4.43%;top:.3em;font-family:Wingd=ings; font-size:75%'>¨</span></span><span style=3D'font-size:88%'>v is at le=vel k-1 and so cannot be a leaf. </span></span></div> <div id=3D"_x0000_s37891_202_251" style=3D'mso-line-spacing:"90 20 0";posi=tion: absolute;top:52.5%;left:5.24%;width:102.05%;height:6.0%;white-space:nowrap='><span style=3D'position:absolute;top:0%;left:3.66%;width:96.33%;height:100.0%'><=span style=3D'font-size:88%'><span style=3D'mso-special-format:bullet;color:#99=99CC; mso-color-index:5;position:absolute;left:-3.8%;top:.3em;font-family:Wingdi=ngs; font-size:75%'>¨</span></span><span style=3D'font-size:88%'>Hence subt=ree rooted at v has height at least 2. </span></span></div> <div id=3D"_x0000_s37891_251_293" style=3D'mso-line-spacing:"90 20 0";posi=tion: absolute;top:59.25%;left:5.24%;width:88.2%;height:6.0%;white-space:nowrap'=><span style=3D'position:absolute;top:0%;left:4.24%;width:95.75%;height:100.0%'><=span style=3D'font-size:88%'><span style=3D'mso-special-format:bullet;color:#99=99CC; mso-color-index:5;position:absolute;left:-4.43%;top:.3em;font-family:Wingd=ings; font-size:75%'>¨</span></span><span style=3D'font-size:88%'>Height-bal=anced property is violated at u </span></span></div> <div style=3D'mso-line-spacing:"90 20 0";position:absolute;top:66.0%;left:=5.24%; width:88.2%;height:6.0%'><span style=3D'position:absolute;top:0%;left:4.24=%; width:95.75%;height:100.0%'><span style=3D'font-size:88%;visibility:hidden='><span style=3D'mso-special-format:bullet;color:#9999CC;mso-color-index:5;positio=n: absolute;left:-4.43%;font-family:Wingdings;font-size:75%'>¨</span></sp=an><span style=3D'font-size:88%;mso-special-format:lastCR;display:none'> </span=></span></div> </div> <v:group id=3D"_x0000_s37903" style=3D'position:absolute;left:456pt;top:35=5.125pt; width:156pt;height:148.875pt' coordorigin=3D"3648,2841" coordsize=3D"1248=,1191"> <v:line id=3D"_x0000_s37896" style=3D'position:absolute' from=3D"3744,355=2" to=3D"4080,3936" strokecolor=3D"black [1]"> <v:shadow color=3D"#00007d [2]"/> </v:line><v:line id=3D"_x0000_s37895" style=3D'position:absolute;flip:x' =from=3D"3744,3072" to=3D"4368,3552" strokecolor=3D"black [1]"> <v:shadow color=3D"#00007d [2]"/> </v:line><v:oval id=3D"_x0000_s37892" style=3D'position:absolute;left:427=2;top:2976; width:192;height:192;mso-wrap-style:none;v-text-anchor:middle' fillcolor==3D"#cccce6 [7]" strokecolor=3D"black [1]"> <v:shadow color=3D"#00007d [2]"/> </v:oval><v:oval id=3D"_x0000_s37893" style=3D'position:absolute;left:364=8;top:3456;
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<body lang=3DEN-US style=3D'margin:0px;background-color:black'onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()"onkeypress=3D"_KPH()" oa:clockstart=3D"onstart">
<div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px;width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%=, 101%, 101%, 0%);visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#00007D,#000000,#9999FF,#9999CC,#666699,#CCCCE6" masterhref=3D"master04.xml"> <p:shaperange href=3D"master04.xml#_x0000_s30721"/><![if !ppt]><p:shaperan=ge href=3D"master04.xml#_x0000_s30723"/> <div v:shape=3D"_x0000_s30723" class=3DO> <div style=3D'text-align:right;mso-char-wrap:1;mso-kinsoku-overflow:1; position:absolute;top:94.0%;left:72.65%;width:21.53%;height:3.5%'><span style=3D'font-family:"Arial Black";font-size:50%;mso-field-code:meta16'>9<=/span><span style=3D'font-family:"Arial Black";font-size:50%;mso-special-format:lastCR; display:none'> </span></div> </div> <p:shaperange href=3D"master04.xml#_x0000_s30724"/><![if !vml]><img border==3D0 v:shapes=3D"_x0000_s30724,_x0000_s30725,_x0000_s30726,_x0000_s30727,_x0000=_s30728,_x0000_s30729,_x0000_s30730,_x0000_s30731,_x0000_s30732,_x0000_s307=33" src=3D"master04_image001.gif" style=3D'position:absolute;top:0%;left:0%; width:100.37%;height:8.5%'><![endif]><p:shaperange href=3D"master04.xml#_x0000_s30736"/><![endif]><p:shaperange href=3D"master04.xml#_x0000_m30734"/><v:shape id=3D"_x0000_s38914" type==3D"#_x0000_m30734" style=3D'position:absolute;left:36pt;top:36pt;width:9in;height:60pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape> <div v:shape=3D"_x0000_s38914" class=3DT style=3D'position:absolute;top:8.=25%; left:5.99%;width:88.2%;height:8.5%'>Structure of an AVL tree (3)</div> <p:shaperange href=3D"master04.xml#_x0000_m30735"/><v:shape id=3D"_x0000_s=38915" type=3D"#_x0000_m30735" style=3D'position:absolute;left:36pt;top:102pt;wi=dth:9in; height:402pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/><p:animation number=3D"2" de=lay=3D"0"
type=3D"text" level=3D"1" effect=3D"cut" direction=3D"noBlack"/></v:shap=e> <div v:shape=3D"_x0000_s38915" class=3DB> <div id=3D"_x0000_s38915_0_50" style=3D'position:absolute;top:20.0%;left:5=.99%; width:97.56%;height:14.0%;white-space:nowrap'><span style=3D'position:abso=lute; top:0%;left:3.83%;width:96.16%;height:48.21%'><span class=3DBB style=3D'po=sition: absolute;left:-3.99%;top:.3em'>¨</span>By previous claim, all levels 1= to k-1 are </span><span style=3D'position:absolute;top:50.0%;left:3.83%;width=:86.56%; height:48.21%'>full. </span></div> <div id=3D"_x0000_s38915_50_86" style=3D'position:absolute;top:35.5%;left:=5.99%; width:88.2%;height:6.75%;white-space:nowrap'><span style=3D'position:absol=ute; top:0%;left:4.24%;width:95.75%;height:100.0%'><span class=3DBB style=3D'po=sition: absolute;left:-4.43%;top:.3em'>¨</span>Hence tree has at least 2<span style=3D'mso-hansi-font-family:Arial;font-size:66%;position:relative;top:-=.45em; mso-text-raise:30%'>k-1</span><span style=3D'mso-spacerun:yes'> </span>nodes. </span></div> <div id=3D"_x0000_s38915_86_151" style=3D'position:absolute;top:44.0%;left=:5.99%; width:100.18%;height:14.0%;white-space:nowrap'><span style=3D'position:abs=olute; top:0%;left:3.73%;width:96.26%;height:48.21%'><span class=3DBB style=3D'po=sition: absolute;left:-3.88%;top:.3em'>¨</span>Since height of tree is at most 2k-1 it has </span><span style=3D'position:absolute;top:51.78%;left:3.73%; width:84.29%;height:48.21%'>at most 2<span style=3D'mso-hansi-font-family:=Arial; font-size:66%;position:relative;top:-.45em;mso-text-raise:30%'>2k-1</span>=<span style=3D'mso-spacerun:yes'> </span>nodes. </span></div> <div id=3D"_x0000_s38915_151_175" style=3D'position:absolute;top:59.75%; left:5.99%;width:88.2%;height:6.75%;white-space:nowrap'><span style=3D'position:absolute;top:0%;left:4.24%;width:95.75%;height:100.0%'><=span class=3DBB style=3D'position:absolute;left:-4.43%;top:.3em'>¨</span>Th=us 2<span style=3D'mso-hansi-font-family:Arial;font-size:66%;position:relative;top:-=.45em; mso-text-raise:30%'>k-1</span><span style=3D'mso-spacerun:yes'> </span><=3D n <=3D 2<span style=3D'=mso-hansi-font-family: Arial;font-size:66%;position:relative;top:-.45em;mso-text-raise:30%'>2k-1<=/span> </span></div> <div id=3D"_x0000_s38915_175_207" style=3D'position:absolute;top:68.25%; left:5.99%;width:88.2%;height:6.75%;white-space:nowrap'><span style=3D'position:absolute;top:0%;left:4.24%;width:95.75%;height:100.0%'><=span class=3DBB style=3D'position:absolute;left:-4.43%;top:.3em'>¨</span>Su=bstituting h for 2k-1 we get </span></div> <div id=3D"_x0000_s38915_207_239" style=3D'position:absolute;top:76.75%; left:5.99%;width:88.2%;height:6.75%;white-space:nowrap'><span style=3D'mso-special-format:nobullet;display:none;color:#9999CC;mso-color-=index: 5;font-family:Wingdings;font-size:75%'>¨</span><span style=3D'mso-spacerun:yes'>  =
; </span>2<span style=3D'mso-hansi-font-family:Arial;font-size:66%;position:=relative; top:-.45em;mso-text-raise:30%'>(h-1)/2</span><span style=3D'mso-spacerun:yes'> </span><=3D n <=3D 2<span style=3D'=mso-hansi-font-family: Arial;font-size:66%;position:relative;top:-.45em;mso-text-raise:30%'>h</sp=an></div> </div></p:slide></div>
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<body lang=3DEN-US style=3D'margin:0px;background-color:black'onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()"onkeypress=3D"_KPH()" oa:clockstart=3D"onstart">
<div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px;width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%=, 101%, 101%, 0%);visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#00007D,#000000,#9999FF,#9999CC,#666699,#CCCCE6" masterhref=3D"master04.xml"> <p:shaperange href=3D"master04.xml#_x0000_s30721"/><![if !ppt]><p:shaperan=ge href=3D"master04.xml#_x0000_s30723"/> <div v:shape=3D"_x0000_s30723" class=3DO> <div style=3D'text-align:right;mso-char-wrap:1;mso-kinsoku-overflow:1; position:absolute;top:94.0%;left:72.65%;width:21.53%;height:3.5%'><span style=3D'font-family:"Arial Black";font-size:50%;mso-field-code:meta16'>10=</span><span style=3D'font-family:"Arial Black";font-size:50%;mso-special-format:lastCR;
display:none'> </span></div> </div> <p:shaperange href=3D"master04.xml#_x0000_s30724"/><![if !vml]><img border==3D0 v:shapes=3D"_x0000_s30724,_x0000_s30725,_x0000_s30726,_x0000_s30727,_x0000=_s30728,_x0000_s30729,_x0000_s30730,_x0000_s30731,_x0000_s30732,_x0000_s307=33" src=3D"master04_image001.gif" style=3D'position:absolute;top:0%;left:0%; width:100.37%;height:8.5%'><![endif]><p:shaperange href=3D"master04.xml#_x0000_s30736"/><![endif]><p:shaperange href=3D"master04.xml#_x0000_m30734"/><v:shape id=3D"_x0000_s39938" type==3D"#_x0000_m30734" style=3D'position:absolute;left:36pt;top:36pt;width:9in;height:66pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape> <div v:shape=3D"_x0000_s39938" class=3DT style=3D'position:absolute;top:8.=75%; left:5.99%;width:91.76%;height:8.5%'>Summary of AVL tree structure</div> <p:shaperange href=3D"master04.xml#_x0000_m30735"/><v:shape id=3D"_x0000_s=39939" type=3D"#_x0000_m30735" style=3D'position:absolute;left:36pt;top:120pt;wi=dth:9in; height:384pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/><p:animation number=3D"2" de=lay=3D"0" type=3D"text" level=3D"1" effect=3D"cut" direction=3D"noBlack"/></v:shap=e> <div v:shape=3D"_x0000_s39939" class=3DB> <div id=3D"_x0000_s39939_0_87" style=3D'position:absolute;top:23.25%;left:=5.99%; width:102.8%;height:14.0%;white-space:nowrap'><span style=3D'position:abso=lute; top:0%;left:3.64%;width:96.35%;height:48.21%'><span class=3DBB style=3D'po=sition: absolute;left:-3.78%;top:.3em'>¨</span>In an AVL tree of height h, the leaf closest </span><span style=3D'position:absolute;top:51.78%;left:3.64%; width:84.15%;height:48.21%'>to the root is at level at least (h+1)/2. =</span></div> <div id=3D"_x0000_s39939_87_155" style=3D'position:absolute;top:39.0%;left=:5.99%; width:103.18%;height:14.0%;white-space:nowrap'><span style=3D'position:abs=olute; top:0%;left:3.62%;width:96.18%;height:48.21%'><span class=3DBB style=3D'po=sition: absolute;left:-3.77%;top:.3em'>¨</span>On the first (h-1)/2 levels the= AVL tree is a </span><span style=3D'position:absolute;top:50.0%;left:3.62%; width:81.85%;height:48.21%'>complete binary tree. </span></div> <div id=3D"_x0000_s39939_155_215" style=3D'position:absolute;top:54.5%;lef=t:5.99%; width:101.87%;height:14.0%;white-space:nowrap'><span style=3D'position:abs=olute; top:0%;left:3.67%;width:96.32%;height:48.21%'><span class=3DBB style=3D'po=sition: absolute;left:-3.81%;top:.3em'>¨</span>After (h-1)/2 levels the AVL tr=ee may start </span><span style=3D'position:absolute;top:51.78%;left:3.67%; width:82.9%;height:48.21%'>“thinning out”. </span></div>
<div id=3D"_x0000_s39939_215_283" style=3D'position:absolute;top:70.25%; left:5.99%;width:92.69%;height:14.0%;white-space:nowrap'><span style=3D'position:absolute;top:0%;left:4.04%;width:95.95%;height:48.21%'><=span class=3DBB style=3D'position:absolute;left:-4.21%;top:.3em'>¨</span>Nu=mber of nodes in the AVL tree is at </span><span style=3D'position:absolute;top:50=.0%; left:4.04%;width:91.11%;height:48.21%'>least 2<span style=3D'mso-hansi-fon=t-family: Arial;font-size:66%;position:relative;top:-.45em;mso-text-raise:30%'>(h-1)=/2</span><span style=3D'mso-spacerun:yes'> </span>and at most 2<span style=3D'mso-ha=nsi-font-family: Arial;font-size:66%;position:relative;top:-.45em;mso-text-raise:30%'>h</sp=an></span></div> </div></p:slide></div>
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<div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px;width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%=, 101%, 101%, 0%);visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#00007D,#000000,#9999FF,#9999CC,#666699,#CCCCE6" masterhref=3D"master04.xml"> <p:shaperange href=3D"master04.xml#_x0000_s30721"/><![if !ppt]><p:shaperan=ge href=3D"master04.xml#_x0000_s30723"/> <div v:shape=3D"_x0000_s30723" class=3DO> <div style=3D'text-align:right;mso-char-wrap:1;mso-kinsoku-overflow:1; position:absolute;top:94.0%;left:72.65%;width:21.53%;height:3.5%'><span style=3D'font-family:"Arial Black";font-size:50%;mso-field-code:meta16'>11=</span><span style=3D'font-family:"Arial Black";font-size:50%;mso-special-format:lastCR; display:none'> </span></div> </div> <p:shaperange href=3D"master04.xml#_x0000_s30724"/><![if !vml]><img border==3D0 v:shapes=3D"_x0000_s30724,_x0000_s30725,_x0000_s30726,_x0000_s30727,_x0000=_s30728,_x0000_s30729,_x0000_s30730,_x0000_s30731,_x0000_s30732,_x0000_s307=33" src=3D"master04_image001.gif" style=3D'position:absolute;top:0%;left:0%; width:100.37%;height:8.5%'><![endif]><p:shaperange href=3D"master04.xml#_x0000_s30736"/><![endif]><p:shaperange href=3D"master04.xml#_x0000_m30734"/><v:shape id=3D"_x0000_s27650" type==3D"#_x0000_m30734" style=3D'position:absolute;left:30pt;top:30pt;width:612pt;height:60pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape> <div v:shape=3D"_x0000_s27650" class=3DT style=3D'position:absolute;top:7.=
25%; left:5.24%;width:83.14%;height:8.5%'><span lang=3DEN-US style=3D'mso-ansi-=language: EN-US'><b>Insertion</b></span><span lang=3DEN-US style=3D'mso-ansi-languag=e:EN-US; mso-special-format:lastCR;display:none'> </span></div> <p:shaperange href=3D"master04.xml#_x0000_m30735"/><v:shape id=3D"_x0000_s=27651" type=3D"#_x0000_m30735" style=3D'position:absolute;left:24pt;top:108pt;wi=dth:678pt; height:354pt' o:userdrawn=3D"f"> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1" size=3D"half"/><p:animation n=umber=3D"2" delay=3D"0" type=3D"text" level=3D"1" effect=3D"dissolve"/><p:tag name==3D"___PPT2001" type=3D"binary" datasize=3D"36" dataoffset=3D"5"/></v:shape> <div v:shape=3D"_x0000_s27651" class=3DHB> <div id=3D"_x0000_s27651_0_108" style=3D'position:absolute;top:21.0%;left:=4.3%; width:109.17%;height:18.5%;white-space:nowrap'><span style=3D'position:abs=olute; top:0%;left:3.43%;width:86.1%;height:32.43%'><span class=3DHBB style=3D'po=sition: absolute;left:-3.98%;top:.3em'>¨</span><span lang=3DEN-US style=3D'mso=-ansi-language: EN-US'>A binary tree T is called </span><span lang=3DEN-US style=3D'color:=#5A45FF; mso-ansi-language:EN-US'>height-balanced</span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'> if for </span></span><span style=3D'pos=ition: absolute;top:33.78%;left:3.43%;width:96.56%;height:32.43%'><span lang=3DEN=-US style=3D'mso-ansi-language:EN-US'>every node v, height of v’s childr=en differ by atmost </span></span><span style=3D'position:absolute;top:67.56%; left:3.43%;width:81.13%;height:32.43%'><span lang=3DEN-US style=3D'mso-ans=i-language: EN-US'>one. </span></span></div> <div id=3D"_x0000_s27651_108_189" style=3D'position:absolute;top:41.0%;lef=t:4.3%; width:99.06%;height:12.25%;white-space:nowrap'><span style=3D'position:abs=olute; top:0%;left:3.78%;width:96.21%;height:48.97%'><span class=3DHBB style=3D'position:absolute;left:-3.92%;top:.3em'>¨</span><span lang=3D=EN-US style=3D'mso-ansi-language:EN-US'>Inserting a node into an AVL tree change=s the </span></span><span style=3D'position:absolute;top:51.02%;left:3.78%;width:89.41%;height:48.97=%'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>heights of some of the node=s in T. </span></span></div> <div id=3D"_x0000_s27651_189_362" style=3D'position:absolute;top:54.75%; left:4.3%;width:109.17%;height:24.75%;white-space:nowrap'><span style=3D'position:absolute;top:0%;left:3.43%;width:90.9%;height:24.24%'><s=pan class=3DHBB style=3D'position:absolute;left:-3.77%;top:.3em'>¨</span><=span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>If insertion causes T to be=come </span><span lang=3DEN-US style=3D'color:#FF3045;mso-ansi-language:EN-US'>unbalanced,</=
span><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'> we </span></span><span style=3D'position:absolute;top:25.25%;left:3.43%;width:96.56%;height:24.24=%'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>travel up the tree from the= newly created node until </span></span><span style=3D'position:absolute;top:49.4=9%; left:3.43%;width:94.68%;height:24.24%'><span lang=3DEN-US style=3D'mso-ans=i-language: EN-US'>we find the first node x such that its grandparent z </span></span>=<span style=3D'position:absolute;top:74.74%;left:3.43%;width:81.13%;height:24.24=%'><span lang=3DEN-US style=3D'mso-ansi-language:EN-US'>is unbalanced node. </s=pan></span></div> <div id=3D"_x0000_s27651_362_393" style=3D'position:absolute;top:80.75%; left:4.3%;width:92.32%;height:6.0%;white-space:nowrap'><span style=3D'posi=tion: absolute;top:0%;left:4.05%;width:95.94%;height:100.0%'><span class=3DHBB style=3D'position:absolute;left:-4.22%;top:.3em'>¨</span><span lang=3D=EN-US style=3D'mso-ansi-language:EN-US'>Let y be the parent of node x.</span></s=pan></div> </div></p:slide></div>
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left:31.88%;width:2.36%;height:7.75%'><span style=3D'mso-hansi-font-family=:Times; font-size:46%;color:black'>5</span><span style=3D'color:#24A63E;mso-specia=l-format: lastCR;display:none'> </span></div> <div v:shape=3D"_x0000_s6333" class=3DO style=3D'position:absolute;top:39.=53%; left:20.86%;width:2.36%;height:7.75%'><span style=3D'mso-hansi-font-family=:Times; font-size:46%;color:black'>1</span><span style=3D'color:#24A63E;mso-specia=l-format: lastCR;display:none'> </span></div> <div v:shape=3D"_x0000_s6334" class=3DO style=3D'position:absolute;top:59.=68%; left:31.1%;width:2.36%;height:7.75%'><span style=3D'mso-hansi-font-family:=Times; font-size:46%;color:black'>1</span><span style=3D'color:#24A63E;mso-specia=l-format: lastCR;display:none'> </span></div> <div v:shape=3D"_x0000_s6335" class=3DO style=3D'position:absolute;top:37.=98%; left:43.3%;width:2.36%;height:7.75%'><span style=3D'mso-hansi-font-family:=Times; font-size:46%;color:red'>3</span><span style=3D'color:#24A63E;mso-special-=format: lastCR;display:none'> </span></div> <div v:shape=3D"_x0000_s6336" class=3DO style=3D'position:absolute;top:17.=82%; left:86.22%;width:2.36%;height:7.75%'><span style=3D'mso-hansi-font-family=:Times; font-size:46%;color:red'>4</span><span style=3D'color:#24A63E;mso-special-=format: lastCR;display:none'> </span></div> <div v:shape=3D"_x0000_s6337" class=3DO style=3D'position:absolute;top:55.=81%; left:65.35%;width:2.36%;height:7.75%'><span style=3D'mso-hansi-font-family=:Times; font-size:46%;color:red'>2</span><span style=3D'color:#24A63E;mso-special-=format: lastCR;display:none'> </span></div> <div v:shape=3D"_x0000_s6338" class=3DO style=3D'position:absolute;top:37.=98%; left:97.63%;width:2.36%;height:7.75%'><span style=3D'mso-hansi-font-family=:Times; font-size:46%;color:black'>1</span><span style=3D'color:#24A63E;mso-specia=l-format: lastCR;display:none'> </span></div> <div v:shape=3D"_x0000_s6352" class=3DO style=3D'position:absolute;top:73.=64%; left:54.33%;width:2.36%;height:7.75%'><span style=3D'mso-hansi-font-family=:Times; font-size:46%;color:black'>1</span><span style=3D'color:#24A63E;mso-specia=l-format: lastCR;display:none'> </span></div> <div v:shape=3D"_x0000_s6359" class=3DO style=3D'position:absolute;top:54.=26%; left:74.01%;width:2.75%;height:10.07%'><span style=3D'mso-hansi-font-famil=y: Times;font-size:63%;color:red'><i>x</i></span><span style=3D'color:#24A63E; mso-special-format:lastCR;display:none'> </span></div> <div v:shape=3D"_x0000_s6360" class=3DO style=3D'position:absolute;top:32.=55%; left:53.54%;width:2.75%;height:10.07%'><span style=3D'mso-hansi-font-famil=y:
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------=_NextPart_01C5151E.C6D72D80Content-Location: file:///C:/D0F41251/lec11_files/script.jsContent-Transfer-Encoding: quoted-printableContent-Type: application/javascript; charset="us-ascii"
function LoadSld(){
var sld=3DGetObj("SlideObj")if( !g_supportsPPTHTML ) {
sld.style.visibility=3D"visible"return
}if( MakeNotesVis() ) returnrunAnimations =3D _InitAnimations();if( IsWin("PPTSld") )
parent.SldUpdated(GetSldId())g_origSz=3DparseInt(SlideObj.style.fontSize)g_origH=3Dsld.style.posHeightg_origW=3Dsld.style.posWidthg_scaleHyperlinks=3D(document.all.tags("AREA").length>0)if( g_scaleHyperlinks )
InitHLinkArray()if( g_scaleInFrame||(IsWin("PPTSld") && parent.IsFullScrMode() ) )
document.body.scroll=3D"no"_RSW()if( IsWin("PPTSld") && parent.IsFullScrMode() )
FullScrInit();=09
MakeSldVis();ChkAutoAdv()
if( runAnimations ){
if( document.all("NSPlay") )document.all("NSPlay").autoStart =3D false;
if( sld.filters && sld.filters.revealtrans )setTimeout( "document.body.start()",
sld.filters.revealtrans.duration * =1000 );
elsedocument.body.start();
}}function MakeSldVis(){
var fTrans=3Dg_showAnimation && SldHasTrans()
if( fTrans ){
if( g_bgSound ) {idx=3Dg_bgSound.indexOf(",");pptSound.src=3Dg_bgSound.substr( 0, idx );pptSound.loop=3D -(parseInt(g_bgSound.substr(idx+1)));
}SlideObj.filters.revealtrans.Apply()
}SlideObj.style.visibility=3D"visible"if( fTrans )
SlideObj.filters.revealtrans.Play()}function MakeNotesVis(){
if( !IsNts() ) return falseSlideObj.style.display=3D"none"nObj =3D document.all.item("NotesObj")parent.SetHasNts(0)if( nObj ) {
nObj.style.display=3D""parent.SetHasNts(1)
}return 1
}function ChkAutoAdv(){
if(SldHasTrans())SlideObj.onfilterchange=3DAutoAdv
elseAutoAdv()
}function AutoAdv(){
if(!IsWin("PPTSld") || !gUseSldTimings )returnvar sld=3DGetCurSld()if( (sld.mAdvDelay>0) && !parent.IsFramesMode() )
setTimeout("parent.GoToNextSld()",sld.mAdvDelay)}function GetObj(id){
if(g_supportsPPTHTML) return document.all(id);else return document.getElementById(id);
}function SldHasTrans() { return SlideObj.style.getAttribute("filter")!=3D""= }function GetSldId() { return sId=3Dlocation.href.substring(location.href.la=stIndexOf('/')+1) }function HideMenu() { if( frames["PPTSld"] && PPTSld.document.all.item("ctx=tmenu") && PPTSld.ctxtmenu.style.display!=3D"none" ) { PPTSld.ctxtmenu.styl=e.display=3D'none'; return true } return false }function IsWin( name ) { return window.name =3D=3D name }function IsNts() { return IsWin("PPTNts") }function IsSldOrNts() { return( IsWin("PPTSld")||IsWin("PPTNts") ) }function SupportsPPTAnimation() { return( navigator.platform =3D=3D "Win32"= && navigator.appVersion.indexOf("Windows")>0 ) }function SupportsPPTHTML(){
var appVer=3Dnavigator.appVersion, msie=3DappVer.indexOf("MSIE "), ver=3D0if( msie >=3D 0 )
ver=3DparseFloat( appVer.substring( msie+5, appVer.indexOf(";",msie) ) )
elsever=3DparseInt(appVer)
return( ver >=3D 4 && msie >=3D 0 )}function _RSW(){
if( !g_supportsPPTHTML || IsNts() || ( !g_scaleInFrame && (!IsWin("PPTSld") || !parent.IsFullScrMode()) ) )
return var padding=3D0; if( IsWin("PPTSld") && parent.IsFramesMode() ) padding=3D6
cltWidth=3Ddocument.body.clientWidth-paddingcltHeight=3Ddocument.body.clientHeight-paddingfactor=3D(1.0*cltWidth)/g_origWif( cltHeight < g_origH*factor )
factor=3D(1.0*cltHeight)/g_origHnewSize =3D g_origSz * factorif( newSize < 1 ) newSize=3D1s=3DSlideObj.styles.fontSize=3DnewSize+"px"s.posWidth=3Dg_origW*factors.posHeight=3Dg_origH*factors.posLeft=3D(cltWidth-s.posWidth+padding)/2s.posTop=3D(cltHeight-s.posHeight+padding)/2if( g_scaleHyperlinks )
ScaleHyperlinks( factor )}function _InitAnimations(){
animRuntimeInstalled =3D ''+document.body.localTime !=3D 'undefined';isFullScreen =3D (window.name =3D=3D "PPTSld") && !parent.IsFramesMode();g_animUseRuntime =3D g_showAnimation && animRuntimeInstalled && !
(isFullSc=reen && parent.IsSldVisited());
if( g_animUseRuntime ) {collSeq =3D document.all.tags("seq");if( collSeq !=3D null ) {
for(ii=3D0;ii<collSeq.length;ii++) {if( collSeq[ii].getAttribute( "p:nodeType" ) =3D=3D
"mainSeq" ) {g_animMainSequence =3D collSeq[ii];break;
}}
}if( g_animItemsToHide ) {
for(jj =3D 0; jj < g_animItemsToHide.length; jj++) {if( hideObj =3D GetObj(g_animItemsToHide[jj]) )
hideObj.runtimeStyle.visibility=3D"hidden";}
}if( g_animInteractiveItems ){
for(jj =3D 0; jj < g_animInteractiveItems.length; jj++) {if( triggerObj =3D GetObj(g_animInteractiveItems[jj]) )
triggerObj.runtimeStyle.cursor=3D"hand";}
}if( gUseSldTimings && ''+g_animSlideTime !=3D 'undefined' ) {
adjustedTime =3D document.body.calculateAutoAdvanceTimes( g_animSlideTim=e, g_animEffectTimings );
if( IsWin("PPTSld") && adjustedTime !=3D g_animSlideTime ) { var sld =3D GetCurSld(); sld.mAdvDelay =3D adjustedTime * 1000;}
}
}return g_animUseRuntime;
}gSldJump =3D 0, gSldJumpTrack =3D 0, gSldJumpIdx =3D "";function _KPH(){
if( IsNts() ) return;if( !parent.IsFramesMode() && event.keyCode =3D=3D 27 && !HideMenu() )
parent.window.close( self );else if( event.keyCode =3D=3D 32 ) {
if( window.name =3D=3D "PPTSld" )parent.PPTSld.DocumentOnClick();
elseparent.GoToNextSld();
}CatchNumKeys( parent, event );
}function CatchNumKeys( win, event ) {
if( win.IsFullScrMode() && (48<=3Devent.keyCode) && (event.keyCode<=3D57) =) {
gSldJump =3D 1;gSldJumpIdx +=3D (event.keyCode-48).toString();
}if( win.IsFullScrMode() && gSldJump && event.keyCode =3D=3D 13 ) {
var numSlds =3D parent.GetSldList().mList.lengthif ( gSldJumpIdx > numSlds )
gSldJumpIdx =3D numSlds;if ( gSldJumpIdx >=3D 0 ) {
if ( gSldJumpIdx =3D=3D 0 )gSldJumpIdx =3D 1;
var jumpTo =3D parseInt(gSldJumpIdx);gSldJump =3D 0; gSldJumpIdx =3D "";win.GoToSld( parent.GetSldList().mList[jumpTo-1].mSldHref )
}}
}function _KDH(){
if( event.keyCode =3D=3D 8 ) {event.returnValue =3D 0;parent.GoToPrevSld();
}}function DocumentOnClick(){
if( IsNts() || parent.HideMenu() ) return;if( ( g_allowAdvOnClick && (window.name=3D=3D"PPTSld") && !
parent.IsFrames=Mode() ) ||
(event && event.keyCode=3D=3D32) ) {if( g_animUseRuntime && g_animMainSequence &&
g_animMainSequence.cangonex=t )
return;parent.GoToNextSld();
}}
var g_supportsPPTHTML =3D SupportsPPTHTML(), g_scaleInFrame =3D 1, gId=3D""=, g_bgSound=3D"", g_scaleHyperlinks =3D false, g_allowAdvOnClick =3D 1, g_showInBrowser ==3D 0, gLoopCont =3D 0, gUseSldTimings =3D 1;var g_showAnimation =3D g_supportsPPTHTML && SupportsPPTAnimation() && ( (w=
indow.name=3D=3D"PPTSld" && !parent.IsFramesMode()) || g_showInBrowser );va=r g_animManager =3D null;var g_animUseRuntime =3D false;var g_animItemsToHide, g_animInteractiveItems, g_animSlideTime;var g_animMainSequence =3D null;var ENDSHOW_MESG=3D"End of slide show, click to exit.", SCREEN_MODE=3D"Fram=es", gIsEndShow=3D0, NUM_VIS_SLDS=3D12, SCRIPT_HREF=3D"script.js", FULLSCR_=HREF=3D"fullscreen.htm";var gCurSld =3D gPrevSld =3D 1, g_offset =3D 0, gNtsOpen =3D gHasNts =3D gO=tlTxtExp =3D 0, gHasNarration =3D 0, gOtlOpen =3D truewindow.gPPTHTML=3DSupportsPPTHTML()var gMainDoc=3Dnew Array(new hrefList("slide0001.htm",1,-1,1),new hrefList(="slide0002.htm",1,-1,1),new hrefList("slide0003.htm",1,-1,1),new hrefList("=slide0024.htm",1,-1,1),new hrefList("slide0027.htm",1,-1,1),new hrefList("s=lide0028.htm",1,-1,1),new hrefList("slide0029.htm",1,-1,1),new hrefList("sl=ide0030.htm",1,-1,1),new hrefList("slide0031.htm",1,-1,1),new hrefList("sli=de0032.htm",1,-1,1),new hrefList("slide0025.htm",1,-1,1),new hrefList("slid=e0005.htm",1,-1,1));function FullScrInit(){
g_allowAdvOnClick =3D GetCurSld().mAdvOnClkdocument.body.style.backgroundColor=3D"black"document.oncontextmenu=3Dparent._CM;document.onkeydown =3D _KDH;document.ondragstart=3DCanceldocument.onselectstart=3DCancelself.focus()
}function Redirect( frmId ){
var str=3Ddocument.location.hash,idx=3Dstr.indexOf('#'), sId=3DGetSldId()if(idx>=3D0) str=3Dstr.substr(1);if( window.name !=3D frmId && ( sId !=3D str) ) {
obj =3D GetObj("Main-File")window.location.href=3Dobj.href+"#"+sIdreturn 1
}return 0
}var MHTMLPrefix =3D CalculateMHTMLPrefix();function CalculateMHTMLPrefix(){
if ( document.location.protocol =3D=3D 'mhtml:') {href=3Dnew String(document.location.href)Start=3Dhref.indexOf('!')+1End=3Dhref.lastIndexOf('/')+1if (End < Start)
return href.substring(0, Start)elsereturn href.substring(0, End)
}return '';
}function GetTags(base,tag){
if(g_supportsPPTHTML) return base.all.tags(tag);else return base.getElementsByTagName(tag);
}function UpdNtsPane(){ if(frames["PPTNts"]) PPTNts.location.replace( MHTMLP=refix+GetHrefObj( gCurSld ).mNtsHref ) }function UpdNavPane( sldIndex ){ if(gNavLoaded) PPTNav.UpdNav() }function UpdOtNavPane(){ if(gOtlNavLoaded) PPTOtlNav.UpdOtlNav() }function UpdOtlPane(){ if(gOtlLoaded) PPTOtl.UpdOtl() }function SetHasNts( fVal )
{if( gHasNts !=3D fVal ) {
gHasNts=3DfValUpdNavPane()
}}function ToggleOtlText(){
gOtlTxtExp=3D!gOtlTxtExpUpdOtlPane()
}function ToggleOtlPane(){
frmset=3Ddocument.all("PPTHorizAdjust")frm=3Ddocument.all("PPTOtl")if( gOtlOpen )
frmset.cols=3D"*,100%"else
frmset.cols=3D"25%,*"gOtlOpen=3D!gOtlOpenfrm.noResize=3D!frm.noResizeUpdOtNavPane()
}function ToggleNtsPane(){
frmset=3Ddocument.all("PPTVertAdjust")frm=3Ddocument.all("PPTNts")if( gNtsOpen )
frmset.rows=3D"100%,*"else
frmset.rows=3D"*,20%"gNtsOpen=3D!gNtsOpenUpdNtsPane()
}function ClearMedia(){
if (PPTSld.pptSound) PPTSld.pptSound.loop =3D 0;}function FullScreen(){
if ( PPTSld.g_animUseRuntime )PPTSld.document.body.pause();
ClearMedia();var href =3D ( document.location.protocol =3D=3D 'mhtml:') ? FULLSCR_HREF
=: FULLSCR_HREF+"#"+GetHrefObj(gCurSld).mSldHref;
if(PPTNav.event.ctrlKey) {var w =3D (window.screen.availWidth * 1.0) / 2.0var h =3D w * (PPTSld.g_origH * 1.0) / PPTSld.g_origWwin =3D
window.open( MHTMLPrefix+href,null,"toolbar=3D0,resizable=3D1,top==3D0,left=3D0," + "width=3D"+ w + ",height=3D" + h );
if( PPTSld.g_animUseRuntime )win.document.body.PPTSldFrameset=3Dwindow;
}else{
win =3D window.open( MHTMLPrefix+href,null,"fullscreen=3Dyes" );if( PPTSld.g_animUseRuntime )
win.document.body.PPTSldFrameset=3Dwindow;}
}function ToggleVNarration(){
rObj=3DPPTSld.document.all("NSPlay")if( rObj && !PPTSld.g_animUseRuntime ) {
if( (rObj.playState =3D=3D 1)||(rObj.playState =3D=3D 0) )rObj.Play()
else if( rObj.playState =3D=3D 2 )rObj.Pause()
elsereturn;
}else if( PPTSld.g_animUseRuntime ){
narObj =3D PPTSld.document.all("narrationID")if( narObj )
narObj.togglePause()}
}function GetCurSldNum(){
obj=3DGetHrefObj(gCurSld)if( obj.mOrigVis =3D=3D 1 )
return obj.mSldIdxelse
return gCurSld}function GetNumSlds(){
if( GetHrefObj(gCurSld).mOrigVis =3D=3D 1 )return GetSldList().mNumVisSlds;
elsereturn GetSldList().mList.length
}function GetSldNum( href ){
for(ii=3D0; ii<GetSldList().mList.length; ii++) {if ( GetSldList().mList[ii].mSldHref =3D=3D href )
return ii+1}return 1
}function GetHrefObj( sldIdx ){ return GetSldList().mList[sldIdx-1] }function IsFramesMode(){ return ( SCREEN_MODE =3D=3D "Frames" ) }function IsFullScrMode(){ return ( SCREEN_MODE =3D=3D "FullScreen" ) }function GoToNextSld(){
ii=3DgCurSld + 1if( GetHrefObj( ii-1 ).mOrigVis =3D=3D 0 ) {
if( ii<=3DGetSldList().mList.length ) {obj=3DGetHrefObj(ii)obj.mVis=3D1GoToSldNum(ii)return
}}else {
obj=3DGetHrefObj( ii )while ( obj && ( obj.mOrigVis =3D=3D 0 ) )
obj=3DGetHrefObj(++ii)if( obj && obj.mOrigVis ) {
GoToSldNum(ii)return
}}if( gSldStack.length > 1 )
PopSldList();
else if( !IsFramesMode() ) { if( gLoopCont )
GoToFirst() else
EndShow()}
}function GoToPrevSld(){
ii=3DgCurSld-1if( ii > 0 ) {
obj=3DGetHrefObj(ii)while ( obj && ( obj.mVis =3D=3D 0 ) && ( ii>0 ) )
obj=3DGetHrefObj(--ii) if( ii =3D=3D 0 ) ii=3D1
GoToSldNum(ii)}
}function GoToFirst(){ GoToSld( GetHrefObj(1).mSldHref ) }function GoToLast(){
ii=3DGetSldList().mList.lengthif( ii !=3D gCurSld )
GoToSld( GetHrefObj(ii).mSldHref )}function GoToSldNum( num ){
if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrueobj =3D GetHrefObj( num )obj.mVis=3D1gPrevSld=3DgCurSldgCurSld =3D num;PPTSld.location.replace(MHTMLPrefix+obj.mSldHref)if( IsFramesMode() ) {
UpdNavPane(); UpdOtlPane(); UpdNtsPane()}
}function GoToSld( href ){
if( PPTSld.event ) PPTSld.event.cancelBubble=3DtrueGetHrefObj( GetSldNum(href) ).mVis=3D1PPTSld.location.replace(MHTMLPrefix+href)
}function SldUpdated( id ){
if( id =3D=3D GetHrefObj(gCurSld).mSldHref ) returngPrevSld=3DgCurSldgCurSld=3DGetSldNum(id)if( IsFramesMode() ) {
UpdNavPane(); UpdOtlPane(); UpdNtsPane()}
}function PrevSldViewed(){ GoToSld( GetHrefObj(gPrevSld).mSldHref ) }function HasPrevSld() { return ( gIsEndShow || ( gCurSld !=3D 1 && GetHrefO=bj( gCurSld-1 ).mVis =3D=3D 1 )||( GetCurSldNum() > 1 ) ) }function HasNextSld() { return (GetCurSldNum() !=3D GetNumSlds()) }function CloseWindow() {
if( HideMenu() ) return;var event =3D PPTSld.event;if( !IsFramesMode() && event && (event.keyCode=3D=3D27 ||
event.keyCode=3D==3D32 || event.type=3D=3D"click" ) )
window.close( self );CatchNumKeys( self, event );
}function Unload() { gIsEndShow=3D0; }function SetupEndShow() {
gIsEndShow=3D1;PPTSld.document.body.scroll=3D"no";PPTSld.document.onkeypress=3DCloseWindow;PPTSld.document.onclick=3DCloseWindow;PPTSld.document.oncontextmenu=3D_CM;
}function EndShow(){
if( IsFramesMode() ) returnif( PPTSld.event ) PPTSld.event.cancelBubble=3Dtruedoc=3DPPTSld.documentdoc.open()doc.writeln('<html><body bgcolor=3Dblack onload=3Dparent.SetupEndShow()
on=unload=3Dparent.Unload()><center><p><font face=3DTahoma color=3Dwhite size==3D2><br><b>' + ENDSHOW_MESG + '</b></font></p></center></body></html>')
doc.close()}function SetSldVisited(){ GetSldList().mList[gCurSld-1].mVisited=3Dtrue }function IsSldVisited(){ return GetSldList().mList[gCurSld-1].mVisited }function hrefList( sldHref, visible, advDelay, advClk ){
this.mSldHref=3D this.mNtsHref =3D sldHrefthis.mOrigVis=3D this.mVis =3D visiblethis.mVisited=3D falsethis.mAdvDelay=3D advDelaythis.mAdvOnClk=3D advClk
}function SldList(arr,curSld,fEnd){
this.mCurSld =3D curSld;this.mList =3D new Array();var idx =3D 1;for(ii=3D0;ii<arr.length;ii++) {
this.mList[ii] =3D new hrefList( arr[ii].mSldHref, arr[ii].mOrigVis, arr[=ii].mAdvDelay, arr[ii].mAdvOnClk );
if( arr[ii].mOrigVis )this.mList[ii].mSldIdx =3D idx++;
}this.mNumVisSlds =3D idx-1;this.fEndShow =3D fEnd;
}function GetSldList() { return gSldStack[gSldStack.length-1] }function GetCurSld() { return parent.GetSldList().mList[parent.gCurSld - 1]= }gSldStack =3D new Array();gSldStack[0] =3D new SldList(gMainDoc,gCurSld,1)function ViewCustomShow(idx,fEnd){
if( !IsFullScrMode() ) return;var sldList =3D new Array();var custShow =3D custShowList[idx-1];var jj =3D 0;for( ii=3D0;ii<custShow.length;ii++ ) {
if( custShow[ii] <=3D gMainDoc.length )sldList[jj++] =3D gMainDoc[custShow[ii]-1];
}if (sldList.length > 0) {PushSldList(sldList,fEnd);gCurSld =3D 1;
}else
if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue}function PushSldList(arr,fEnd) {
var ii =3D gSldStack.length;gSldStack[ii] =3D new SldList(arr,gCurSld,fEnd);GoToSld( gSldStack[ii].mList[0].mSldHref );
}function PopSldList() {
if (gSldStack[gSldStack.length-1].fEndShow)EndShow()
else {gCurSld =3D gSldStack[gSldStack.length-1].mCurSld;gSldStack[gSldStack.length-1] =3D null;gSldStack.length--;var sldList =3D gSldStack[gSldStack.length-1];GoToSld( sldList.mList[gCurSld - 1].mSldHref );}
}var custShowList=3Dnew Array();function ImgBtn( oId,bId,w,action ){
var t=3Dthist.Perform =3D _IBPt.SetActive =3D _IBSetAt.SetInactive=3D _IBSetIt.SetPressed =3D _IBSetPt.SetDisabled=3D _IBSetDt.Enabled =3D _IBSetEt.ChangeIcon =3D nullt.UserAction =3D actiont.ChgState =3D _IBUIt.mObjId =3D oIdt.mBorderId=3D bIdt.mWidth =3D wt.mIsOn =3D t.mCurState =3D 0
}function _IBSetA(){
if( this.mIsOn ) {obj=3Dthis.ChgState( gHiliteClr,gShadowClr,2 )obj.style.posTop=3D0
}}function _IBSetI(){
if( this.mIsOn ) {obj=3Dthis.ChgState( gFaceClr,gFaceClr,1 )obj.style.posTop=3D0
}}function _IBSetP(){
if( this.mIsOn ) {obj=3Dthis.ChgState( gShadowClr,gHiliteClr,2 )obj.style.posLeft+=3D1; obj.style.posTop+=3D1
}}function _IBSetD(){
obj=3Dthis.ChgState( gFaceClr,gFaceClr,0 )obj.style.posTop=3D0
}
function _IBSetE( state ){
var t=3DthisGetObj( t.mBorderId ).style.visibility=3D"visible"if( state !=3D t.mIsOn ) {
t.mIsOn=3Dstateif( state )
t.SetInactive()else
t.SetDisabled()}
}function _IBP(){
var t=3Dthisif( t.mIsOn ) {
if( t.UserAction !=3D null )t.UserAction()
if( t.ChangeIcon ) {obj=3DGetObj(t.mObjId)if( t.ChangeIcon() )
obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-4)*t.mWidth
elseobj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-
0)*t.mWidth}t.SetActive()
}}function _IBUI( clr1,clr2,nextState ){
var t=3DthisSetBorder( GetObj( t.mBorderId ),clr1,clr2 )obj=3DGetObj( t.mObjId )obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-nextState)*t.mWidth-
obj=.style.posTop
t.mCurState=3DnextStatereturn obj
}function TxtBtn( oId,oeId,action,chkState ){
var t=3Dthist.Perform =3D _TBPt.SetActive =3D _TBSetAt.SetInactive=3D _TBSetIt.SetPressed =3D _TBSetPt.SetDisabled=3D _TBSetDt.SetEnabled =3D _TBSetEt.GetState =3D chkStatet.UserAction =3D actiont.ChgState =3D _TBUIt.mObjId =3D oIdt.m_elementsId=3D oeIdt.mIsOn =3D 1
}function _TBSetA(){
var t=3Dthisif( t.mIsOn && !t.GetState() )
t.ChgState( gHiliteClr,gShadowClr,0,0 )}function _TBSetI()
{var t=3Dthisif( t.mIsOn && !t.GetState() )
t.ChgState( gFaceClr,gFaceClr,0,0 )}function _TBSetP(){
if( this.mIsOn )this.ChgState( gShadowClr,gHiliteClr,1,1 )
}function _TBSetD(){
this.ChgState( gFaceClr,gFaceClr,0,0 )this.mIsOn =3D 0
}function _TBSetE(){
var t=3Dthisif( !t.GetState() )
t.ChgState( gFaceClr,gFaceClr,0,0 )else
t.ChgState( gShadowClr,gHiliteClr,1,1 )t.mIsOn =3D 1
}function _TBP(){
var t=3Dthisif( t.mIsOn ) {
if( t.UserAction !=3D null )t.UserAction()
if( !t.GetState ) return
if( t.GetState() )t.SetPressed()
elset.SetActive()
}}function _TBUI( clr1,clr2,lOffset,tOffset ){
SetBorder( GetObj( this.mObjId ),clr1,clr2 )Offset( GetObj( this.m_elementsId ),lOffset,tOffset )
}function Offset( obj, top, left ){ obj.style.top=3Dtop; obj.style.left=3Dle=ft }function SetBorder( obj, upperLeft, lowerRight ){
s=3Dobj.style;s.borderStyle =3D "solid"s.borderWidth =3D 1s.borderLeftColor =3D s.borderTopColor =3D upperLefts.borderBottomColor=3D s.borderRightColor =3D lowerRight
}function GetBtnObj(){ return gBtnArr[window.event.srcElement.id] }function BtnOnOver(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetActive() }function BtnOnDown(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetPressed() }function BtnOnOut(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetInactive() }function BtnOnUp(){
b=3DGetBtnObj()if( b !=3D null )
b.Perform()else
Upd()
}function GetNtsState(){ return parent.gNtsOpen }function GetOtlState(){ return parent.gOtlOpen }function GetOtlTxtState(){ return parent.gOtlTxtExp }function NtsBtnSetFlag( fVal ){
s=3Ddocument.all.item( this.m_flagId ).styles.display=3D"none"if( fVal )
s.display=3D""else
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BuildMenu();var doc=3DPPTSld.document.body,x=3DPPTSld.event.clientX+doc.scrollLeft,y=
=3DPPTSld.event.clientY+doc.scrollTopm =3D PPTSld.document.all.item("ctxtmenu")m.style.pixelLeft=3Dxif( (x+m.scrollWidth > doc.clientWidth)&&(x-m.scrollWidth > 0) )
m.style.pixelLeft=3Dx-m.scrollWidthm.style.pixelTop=3Dyif( (y+m.scrollHeight > doc.clientHeight)&&(y-m.scrollHeight > 0) )
m.style.pixelTop=3Dy-m.scrollHeightm.style.display=3D""
}function _CM()
{if( !parent.IsFullScrMode() ) return;if(!PPTSld.event.ctrlKey) {
ShowMenu()return false
} elseHideMenu()
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if( PPTSld.document.all.item("ctxtmenu") ) returnvar mObj=3DCreateItem( PPTSld.document.body )mObj.id=3D"ctxtmenu"mObj.style.visibility=3D"hidden"var s=3DmObj.styles.position=3D"absolute"s.cursor=3D"default"s.width=3D"120px"SetCMBorder(mObj,"menu","black")var iObj=3DCreateItem( mObj )SetCMBorder( iObj, "threedhighlight","threedshadow" )iObj.style.padding=3D2CreateMenuItem( iObj,sNext,M_GoNextSld,M_True )CreateMenuItem( iObj,sPrev,M_GoPrevSld,M_HasPrevSld )CreateSeparator( iObj )CreateMenuItem( iObj,sEnd,M_End,M_True )mObj.style.visibility=3D"visible"
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e=3DPPTSld.event.srcElementif( e.type=3D=3D"menuitem" && e.IsActive() )
e.Action()else
PPTSld.event.cancelBubble=3Dtrue}function ChangeClr( bg,clr ){
e=3DPPTSld.event.srcElementif( e.type=3D=3D"menuitem" && e.IsActive() ) {
e.style.backgroundColor=3Dbge.style.color=3Dclr
}}function M_HasPrevSld() { return( parent.HasPrevSld() ) }function M_GoNextSld() { if( gIsEndShow ) M_End(); else GoToNextSld() }function M_GoPrevSld() { if( gIsEndShow ) { history.back(); PPTSld.event.ca=ncelBubble=3Dtrue; } else GoToPrevSld() }function M_True() { return true }function M_End() { window.close( self ) }function CreateMenuItem( node,text,action,eval ){
var e=3DCreateItem( node )e.type=3D"menuitem"e.Action=3Dactione.IsActive=3Devale.innerHTML=3Dtextif( !e.IsActive() )
e.style.color=3D"threedshadow"e.onclick=3DPerform
e.onmouseover=3DHighlighte.onmouseout=3DDeselects=3De.style;s.fontFamily=3DsFonts.fontSize=3D"9pt"s.paddingLeft=3D2
}function CreateSeparator( node ){
var sObj=3DCreateItem( node )SetCMBorder(sObj,"menu","menu")var s=3DsObj.styles.borderTopColor=3D"threedshadow"s.borderBottomColor=3D"threedhighlight"s.height=3D1s.fontSize=3D"0px"
}function CreateItem( node ){
var elem=3DPPTSld.document.createElement("DIV")node.insertBefore( elem )return elem
}function SetCMBorder( o,ltClr,rbClr ){
var s=3Do.styles.backgroundColor=3D"menu"s.borderStyle=3D"solid"s.borderWidth=3D1s.borderColor=3DltClr+" "+rbClr+" "+rbClr+" "+ltClr
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<table width=3D"100%" style=3D'font-family:Arial;font-size:9pt'> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(=this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>1</font></div> </td> <td width=3D"100%"> <div id=3DPPTP1 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0001.htm');" id=3DPPTL1>AVL Trees</a></f=ont></div> <div id=3DPPTC1 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>AVL Trees</li> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(=this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>2</font></div> </td> <td> <div id=3DPPTP2 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0002.htm');" id=3DPPTL2>AVL Tree</a></fo=nt></div> <div id=3DPPTC2 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>AVL trees are balanced.</li> <li>An AVL Tree is a binary search tree such that for every internal nod=e v of T, the heights of the children of v can differ by at most 1.</li> <br> <br> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(=this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>3</font></div> </td>
<td> <div id=3DPPTP3 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0003.htm');" id=3DPPTL3>Height of an AVL= Tree</a></font></div> <div id=3DPPTC3 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Proposition: The height of an AVL tree T storing n keys is O(log n).=</li> <li>Justification: The easiest way to approach this problem is to find n=(h): the minimum number of nodes in an AVL tree of height h.</li> <li>We see that n(1) =3D 1 and n(2) =3D 2</li> <li>For h ≥ 3, an AVL tree of height h contains the root node, one= AVL subtree of height h-1 and the other AVL subtree of height h-1 or h-2=.</li> <li>i.e. n(h) =3D 1 + n(h-1) + n(h-2)</li> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(=this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>4</font></div> </td> <td> <div id=3DPPTP4 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0024.htm');" id=3DPPTL4>Height of an AVL= Tree (2)</a></font></div> <div id=3DPPTC4 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Knowing n(h-1) >=3D n(h-2), we get</li> <li><span style=3D'mso-spacerun:yes'> </span>n(h=) =3D n(h-1) + n(h-2) + 1 > 2n(h-2)</li> <ul> <li>n(h) > 2n(h-2)</li> <li><span style=3D'mso-spacerun:yes'> &nbs=p; </span>> 4n(h-4)</li> <li><span style=3D'mso-spacerun:yes'> &nbs=p; </span>> 8n(h-6)</li> <li><span style=3D'mso-spacerun:yes'> &nbs=p; </span>…</li> <li><span style=3D'mso-spacerun:yes'> &nbs=p; </span>> 2<sup>i</sup>n(h-2i)</li> </ul> <li>When i =3D h/2-1 we get: n(h) > 2<sup>h/2-1</sup>n(2) =3D 2<sup>h=/2</sup></li> <li>Taking logarithms: h < 2log n(h)</li> <li>Thus the height of an AVL tree is O(log n)</li> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(=this)" style=3D'cursor:hand'>
<td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>5</font></div> </td> <td> <div id=3DPPTP5 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0027.htm');" id=3DPPTL5>A sharper bound<=/a></font></div> <div id=3DPPTC5 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>We will show how to obtain a sharper bound on the height of an AVL t=ree.</li> <li>We prove using induction that the minimum number of nodes in an AVL =tree of height h, n(h) >=3D c<sup>h</sup>,<span style=3D'mso-spacerun:yes'> </span>where c is some number >=1.</li> <li>Base case: h=3D1. Now n(h) >=3D c > 1.</li> <li>Suppose claim is true for all h < k</li> <li>We have to show that n(k) >=3D c<sup>k</sup></li> <br> <br> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(=this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>6</font></div> </td> <td> <div id=3DPPTP6 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0028.htm');" id=3DPPTL6>Sharper bound (2=)</a></font></div> <div id=3DPPTC6 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>n(k) =3D n(k-1)+n(k-2)+1</li> <li><span style=3D'mso-spacerun:yes'>  =; </span>>=3D c<sup>k-1</sup> + c<sup>k-2</sup><span style=3D'mso-spacerun:yes'> </span>(by induction hypothesis)</=li> <li>We will be able to show that n(k) >=3D c<sup>k</sup> if we can sh=ow that c<sup>k-1</sup> + c<sup>k-2</sup> >=3D c<sup>k</sup>.</li> <li>So c should be such that c<sup>2</sup>-c-1 <=3D 0.</li> <li>The quadratic equation c<sup>2</sup>-c-1=3D0 has</li> <li><span style=3D'mso-spacerun:yes'> </span>roots<span style=3D'mso-spacerun:yes'>  =; </span>and<span style=3D'mso-spacerun:yes'>  =; </span>.</li> <li>Hence we can take c as<span style=3D'mso-spacerun:yes'>  =; </span>which is roughly 1.63</li> <li>Hence AVL tree on n nodes has height atmost log<sub>1.63</sub> n</li> </ul> </font></div> </td>
</tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(=this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>7</font></div> </td> <td> <div id=3DPPTP7 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0029.htm');" id=3DPPTL7>Structure of an =AVL tree</a></font></div> <div id=3DPPTC7 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Consider an AVL tree on n nodes.</li> <li>Consider a leaf which is closest to the root.</li> <li>Suppose this leaf is at level k.</li> <li>We will show that the height of the tree is at most 2k-1.</li> <br> <br> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(=this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>8</font></div> </td> <td> <div id=3DPPTP8 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0030.htm');" id=3DPPTL8>Structure of an =AVL tree (2)</a></font></div> <div id=3DPPTC8 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Claim: Since closest leaf is at level k all nodes at levels 1..k-2 h=ave 2 children.</li> <li>Proof is by contradiction</li> <li>Suppose node u at level k-2 has only 1 child, v.</li> <li>v is at level k-1 and so cannot be a leaf.</li> <li>Hence subtree rooted at v has height at least 2.</li> <li>Height-balanced property is violated at u</li> <br> <br> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(=this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>9</font></div> </td> <td> <div id=3DPPTP9 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0031.htm');" id=3DPPTL9>Structure of an =AVL tree (3)</a></font></div> <div id=3DPPTC9 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>By previous claim, all levels 1 to k-1 are full.</li>
<li>Hence tree has at least 2<sup>k-1</sup> nodes.</li> <li>Since height of tree is at most 2k-1 it has at most 2<sup>2k-1</sup> nodes.</li> <li>Thus 2<sup>k-1</sup> <=3D n <=3D 2<sup>2k-1</sup></li> <li>Substituting h for 2k-1 we get</li> <li><span style=3D'mso-spacerun:yes'>  =; </span>2<sup>(h-1)/2</sup> <=3D n <=3D 2<sup>h</sup></li> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(=this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>10</font></div> </td> <td> <div id=3DPPTP10 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0032.htm');" id=3DPPTL10>Summary of AVL =tree structure</a></font></div> <div id=3DPPTC10 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>In an AVL tree of height h, the leaf closest to the root is at level= at least (h+1)/2.</li> <li>On the first (h-1)/2 levels the AVL tree is a complete binary tree.<=/li> <li>After (h-1)/2 levels the AVL tree may start “thinning out̶=1;.</li> <li>Number of nodes in the AVL tree is at least 2<sup>(h-1)/2</sup> and =at most 2<sup>h</sup></li> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(=this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>11</font></div> </td> <td> <div id=3DPPTP11 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0025.htm');" id=3DPPTL11>Insertion</a></=font></div> <div id=3DPPTC11 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>A binary tree T is called height-balanced if for every node v, heigh=t of v’s children differ by atmost one.</li> <li>Inserting a node into an AVL tree changes the heights of some of the nodes in T.</li> <li>If insertion causes T to become unbalanced, we travel up the tree fr=om the newly created node until we find the first node x such that its grandparent z is unbalanced node.</li> <li>Let y be the parent of node x.</li> </ul> </font></div>
</td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(=this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>12</font></div> </td> <td> <div id=3DPPTP12 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0005.htm');" id=3DPPTL12>Insertion (2)</=a></font></div> </td> </tr></table>
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