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Binary Search TreesBinary Search Trees
Lecture 5
1
Binary search tree sortBinary search tree sort
2
Binary-search-tree propertyBinary-search-tree property
xyykeyxkey
xyxkeyykey
x
of subtreeright in the node a is if
of subtereeleft in the node a is if
esearch trebinary ain node a be Let
3
Binary-search-tree propertyBinary-search-tree property
• Binary Search Tree can be implemented as a linked data structure in which each node is an object with three pointer fields.
• The three pointer fields left, right and p point to the nodes corresponding to the left child, right child and the parent respectively.
• NIL in any pointer field signifies that there exists no corresponding child or parent.
• The root node is the only node in the BTS structure with
NIL in its p field.
4
Binary-search-tree propertyBinary-search-tree property
5
Inorder-tree walkInorder-tree walk
)()(
)WALK(-TREE-INORDER 4
print 3
)WALK(-TREE-INORDER 2
NIL 1
)WALK(-TREE-INORDER
nnT
xright
xkey
xleft
x
x
thenif
During this type of walk, we visit the root of a subtree between the left subtree visit and right subtree visit.
6
Preorder-tree walkPreorder-tree walk
)()(
)WALK(-TREE-DER PREOR 4
)WALK(-TREE-DER PREOR 3
print 2
NIL 1
)WALK(-TREE-PREORDER
nnT
xright
xleft
xkey
x
x
thenif
In this case we visit the root node before the nodes in either subtree.
7
PPostostorder-tree walkorder-tree walk
)()(
print 4
)WALK(-TREE-RDER POSTO 3
)WALK(-TREE-RDER POSTO 2
NIL 1
)WALK(-TREE-POSTORDER
nnT
xkey
xright
xleft
x
x
thenif
We visit the root node after the nodes in its subtrees.
8
Sorting by binary-search-treeSorting by binary-search-tree 6
3 7
2 5 8
12
6
3
12
1 NIL
3 PRINT 24
NIL
3 PRINT 3
41 2 NIL3 PRINT 54 NIL
2
1 2 NIL3 PRINT 84 NIL
5
3 PRINT 64
1 2 NIL3 PRINT 74
12
77
8
)WALK(-TREE-INORDER 4
print 3
)WALK(-TREE-INORDER 2
NIL 1
)WALK(-TREE-INORDER
xright
xkey
xleft
x
x
then
if
Sorting by binary-search-tree Sorting by binary-search-tree
WALK-TREE-INORDER execute to timethe reflects where
)1()()(
Then nodes.
1 has right tree whoseand nodes has left tree whose
node aon called is WALK-TREE-INORDERthat Suppose
d
dknTkTnT
knk
x
10
Sorting by binary-search-treeSorting by binary-search-tree (contnd.) (contnd.)
)(
)(
)()(
))1)((())((
)1()()(
)()( that providingby methodon substituti Use
n
cndc
dcdccndc
dckndcckdc
dknTkTnT
cndcnT
11
Searching for a keySearching for a key
15
6 18
3
2 4
7 17
13
20
9
Search for 13
12
Searching for a keySearching for a key
),SEARCH(-TREE 5
),SEARCH(-TREE 4
3
2
or NILL 1
),SEARCH(-TREE
kxright
kxleft
xkeyk
x
xkeykeyx
kx
returnelse
returnthen
if
returnthen
if
)()( hOnT 13
Searching for minimumSearching for minimum
15
6 18
3
2 4
7 17
13
20
9
Minimum
14
Searching for minimumSearching for minimum
x
xleftx
xleft
x
return
do
while
3
][ 2
NILL ][ 1
)MINIMUM(-TREE
)()( hOnT 15
Searching for maximumSearching for maximum
15
6 18
3
2 4
7 17
13
20
9
Maximum
16
Searching for maximumSearching for maximum
x
xrightx
xright
x
return
do
while
3
][ 2
NILL ][ 1
)MAXIMUM(-TREE
)()( hOnT 17
Searching for successorSearching for successor
The successor of a node x is the node with the smallest key greather than key[x]
18
Searching for successorSearching for successor
15
6 18
3
2 4
7 17
13
20
9
Successor of 15
Case 1: The right subtree of node x is nonempty 19
Searching for successorSearching for successor
15
6 18
3
2 4
7 17
13
20
9
Successor of 13
Case 2: The right subtree of node x is empty 20
Searching for successorSearching for successor
15
6 18
3
2 4
7 17
13
20
9
Successor of 4search untily.left = x
Case 2: The right subtree of node x is empty 21
Searching for successorSearching for successor
y
ypy
yx
yrightxy
xpy
xright
xright
x
return
do
while
returnthen
if
7
][ 6
5
][ and NILL 4
][ 3
])[(MINIMUMTREE 2
NILL ][ 1
)SUCCESSOR(-TREE
)()( hOnT 22
Go upper until y.left = x
InsertionInsertion
12
5 18
2
13 17
9 15 19
Inserting an item with key 13 23
InsertionInsertion
][ 13
][ 12
][][ 11
empty was Tree ][ 10
NILL 9
][ 8
][ 7
][ 6
][][ 5
4
NILL 3
][ 2
NILL 1
),(INSERTTREE
zyright
zyleft
ykeyzkey
TzTroot
y
yzp
xrightx
xleftx
xkeyzkey
xy
x
Trootx
y
zT
else
then
ifelse
then
if
else
then
if
do
while
)()( hOnT
24
DeletionDeletion
15
5 16
2
18
12
23
20
Deleting an item with key 13 (z has no children)
10 13 z
5
3
6
7
25
DeletionDeletion
15
5 16
2
18
12
23
20
Deleting an item with key 13 (z has no children)
10
5
3
26
6
7
DeletionDeletion
15
5 16 z
2
18
12
23
20
Deleting a node with key 16 (z has only one child )
10 13
5
3
27
6
7
DeletionDeletion
15
5
2 12
Deleting a node with key 16 (z has only one child )
10 13
5
318 23
20
28
6
7
DeletionDeletion
15
5 16
2
18
12
23
20
Deleting a node with key 5 (6 successor of 5) (z has two children)
10
z 5
3
6 y
7
13
29
DeletionDeletion
15
5 16
2
18
12
23
20
Deleting a node with key 5 (z has two children)
10
z 5
3
6 y
7
30
13
DeletionDeletion
15
5 16
2
18
12
23
20
Deleting a node with key 5 (z has two children)
10
6
3
7
31
13
DeletionDeletion
17
into data satellite s'copy 16
][][ 15
14
]][[ 13
]][[ 12
]][[ 11
][ 10
NIL][ 9
][][ 8
NIL 7
][ 6
][ 5
NIL][ 4
)SUCCESSOR(-TREE 3
2
NIL][or NIL][ 1
),(DELETETREE
return
then
if
else
then
ifelse
then
if
then
if
else
then
if
else
then
if
zy
ykeyzkey
zy
xypright
xypleft
yplefty
xTroot
yp
ypxp
x
yrightx
yleftx
yleft
zy
zy
zrightzleft
zT
)()( hOnT
32
