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UNIT III
Trees
2
SyllabusDifference in linear and non-linear data structure
Trees and binary trees-concept and terminology
Binary tree as an ADT.
Algorithm for tree traversals (recursive and non recursive).
Conversion of general tree to binary tree.
Binary search trees,
Concept of threaded binary tree.
Threaded binary tree as an ADT.
Preorder, Inorder traversals of inorder threaded binary tree
Applications of trees
3
Non Linear DSLinear Data Structure :one to one relationship
To store info within the computer system having
a root directory , subdirectory & related fields
Linear DS: more space & time
Non linear DS: more complex relationship
4
Difference in Linear and Non-linear Data Structures
LINEAR DATA STRUCTURES NON-LINEAR DATA STRUCTURES
Linear Data structures are used to represent sequential data.
Non-linear data structures are used to represent hierarchical data.
Linear data structures are easy to implement These data structures are difficult to implement.
Implementation: Linear data structures are implemented using array and linked lists
Implementation: Non-linear data structures are mostly implemented using linked lists.
e.g: The basic linear data structures are list, stack and queue.
e.g: The basic non-linear data structures are trees and graphs.
For the implementation of linear data structures, we don’t need non-linear data structures.
For the implementation of non-linear data structures, we need linear data structures.
USE: These are mostly used in application software development.
USE: These are used for the development of game theory, artificial intelligence, image processing
6/1/2014
5
Concept of tree data structureTree data structure Definition: A tree is a finite set of one or more nodes
such that 1. There is a specially designated node called the root.
2. The remaining nodes are partitioned into n>=0 disjoint sets T1,….,Tn, where each of these sets is a tree. T1,…,Tn are called the subtrees of the root.
A
F
B C D
E
6
Tree TerminologyNode: information
Edges: branches
Root A root is always at the top of
the tree. The root is A.
Parent It is immediate predecessor of
node. A is parent of B,C,D
Child It is immediate successors of
node. B,C,D are children of A.
A
F
B C D
E
7
Tree TerminologyLeaf Node
Siblings Nodes with same parent are
sibling. B,C,D are siblings
Descendent
Ancestors
Path It is no of successive edges
from sour to dest. Path from A to E is A-B,B-E
A
F
B C D
E
8
Degree of Node
The degree of a node is a no. of children of a node.
Leaf node is terminal node & has
no children.Degree 2
Degree 0
9
Height (or depth) of a tree The depth of a tree is the maximum depth of any of its leaves
tree depth = 3
tree depth = 2
tree depth = 0
Root
Root
10
ForestCollection of trees is known as Forest
When we remove root from tree, we get Forest
Original Tree
Root
Root removed
New roots
Forest
11
Binary TreeA tree is binary if each node of the tree can have maximum of two children
Outgoing Degree : 0 , 1 , 2
Its Empty tree or finite set of nodes
Subtrees: Left & Right subtree
D
A
F
B
E E
12
13
Left skewed binary tree Right skewed binary tree Complete binary tree Strictly binary tree Full binary tree
Types of Binary Trees
14
LEFT SKEWED
Every intermediate node is the left child of its root
A
C
D
B
15
RIGHT SKEWED
Every intermediate node is the right child of its root
A
B
C
D
16
Strictly Binary Tree• If every non leaf node in a binary tree has non empty left
and right subtrees, the tree is termed as a strictly binary tree. A node will have either two children or no child at all
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• Example :• Expression
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• binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible
• A complete binary tree is define as binary tree where All of its leaves or terminal nodes are on level (n-1) or n Leaves are filled from left to right
Complete Binary Tree
19
• Every node has two children, so that there are 2d nodes at level d.
Full Binary Tree
20
A
E
B C
D
1
32
4 5 F7
0 1 2 3 4 5 6 7
7 A B C D E \0 F
Binary Tree representation using Array
No of Nodes : Index 0Root Node : Index 1Left Child: Parent Index * 2Right Child: Parent Index*2 + 1
21
22
23
Problems of array representation inefficient storage utilization ideal for complete binary trees hard to insert/delete
24
Binary Tree using Linked List
Linked Representation of binary tree is more efficient than array representation
typedef struct node
{
int data;
struct node *left,*right
};
left_child data right_child
left_child right_child
data
25
Example: A
C
H
E F
I
GD
B
A
B C
0 E 0 0 F 0D 0 G 0
0 I 00 H 0
root
26
Implementationclass Tree{ struct Node{int Data;Node *Left,*Right};Node * Root;public:Tree() { Root=NULL;}void Insert();void Display();}
27
void Tree::Insert( int Num){ Node *nNode=new Node;
nNode->Data=Num;nNode->Left=nNode->Right=NULL;While( 1 ) { cout<<“Enter Direction( L/R)”; cin>>Direction; if(Direction== ‘L’) { if(Temp->Left== NULL) { Temp->Left=nNode; break;} else Temp=Temp->Left; }//
else//insert at right { if(Temp->Right== NULL) { Temp->Right=nNode; break;} else Temp=Temp->Right; }}//while ( 1 )}//end of Insert
28
Binary tree traversalsvisit each node in the tree exactly oncewhat order?
inorder: LVR (Left Visit Right) preorder: VLR (Visit Left Right) postorder: LRV (Left Right Visit)
29
Recursivevoid inorderRec( Node *T)
{
if(T!=NULL)
{ inorderR(T->Left);
cout<<T->data;
inorderR(T->Right);
}
}
7/1/2014
30
How system stack work?void inorderRec(Node *T)
{
1001: if(T!=NULL)
1002: {
1003: inorderR(T->Left);
1004: cout<<T->data;
1005: inorderR(T->Right);
1006: }
}
31
Recursivevoid preorderR(node *T)
{
if(T!=NULL)
{
cout<<T->data;
preorderR(T->Left);
preorderR(T->Right);
}
}
8/1/2014
32
Recursivevoid postorderR(node *T)
{
if(T!=NULL)
{
postorderR(T->Left);
postorderR(T->Right);
cout<<T->data;
}
}
33
Expression Tree1. A+B*C
2. (A+B)*C
3. A+B+C
4. A*(B+C)-D/(E-F)
5. A*B+C-D/F
6. A+(B-C)*D$(E*F)
7. (A+B*C)$((A+B)*C)
8. (a+(b-c))*((d-e)/(f+g-h))
34
Convert Postfix Expression to Binary Tree
1. ABC*+
2. AB+C*
3. AB+C+
4. ABC+*DEF-/-
5. AB* C+DF/-
6. ABC-DEF*$*+
7. ABC*+ AB+C*$
8. abc-+ de-fg+h-/*
35
Convert Prefix Expression to Binary Tree
1. +A*BC
2. *+ABC
3. ++ABC
4. -*A+BC/D-EF
5. -+*ABC/DF
6. +A*-BC$D*EF
7. $+A*BC*+ABC
8. *+a-bc/-de-+fgh
36
struct Tree
{ char Data; struct Tree *Left,*Right;
Tree(char D='\0'){ Data=D; Left=Right=NULL;}
};//Structure for Expression
class Stack
{
Tree *Stk[20]; int Top;
public:
Stack(){Top=-1;}
int isEmpty();
int isFull();
void Push(Tree*);
Tree* Pop();
};
37
Tree *Create()
{ char Estr[25]; int I=0; Stack S;
cout<<"Enter Postfix Expression“; cin>>Estr;
while(Estr[I]!='\0')
{
Tree *Node=new Tree(Estr[I]);
if(isalnum(Estr[I])) S.Push(Node);
else
{ Node->Right=S.Pop();
Node->Left=S.Pop();
S.Push(Node);
}
I++;
}
return S.Pop(); }
38
9/1/2014
39
Non-recursive Inordervoid NonRin(Tree *Root)
{
Stack S;
while(!S.isEmpty()||Root!=NULL)
{
while(Root!=NULL)
{
S.Push(Root);
Root=Root->Left;
}
Root=S.Pop();
cout<<Root->Data;
Root=Root->Right;
}
}
40
Non-recursive Preordervoid NonRpre(Tree *Root)
{
Stack S;
while(!S.isEmpty()||Root!=NULL)
{
while(Root!=NULL)
{
cout<<Root->Data;
S.Push(Root);
Root=Root->Left;
}
Root=S.Pop();
Root=Root->Right;
}
}
41
Non-recursive Postordervoid NonRpost(Tree *Root)
{
Stack S; int I=0; char Str[25];
while(!S.isEmpty()||Root!=NULL)
{
while(Root!=NULL)
{
Str[I++]=Root->Data;
S.Push(Root);
Root=Root->Right;
}
Root=S.Pop();
Root=Root->Left;
}
Str[I]='\0';
cout<<strrev(Str);
}
42
A General Tree
Convert general tree into Binary tree(leftmost child right sibling relation)
Convert forest into binary tree
13/1/2014
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Example :
K
E F G H JI
B DC
A
L M
K
E F G H JI
B DC
A
L M
K
E
F G
H
J
I
B
D
C
A
L
M
Fig: Tree
left-child right-child tree representation of a tree
Binary tree
44
Create Binary tree1. Inorder: BAC Preorder:ABC
2. Inorder: DBEAFHG Preorder:ABDEHFG
3. Inorder:BACDEHFG Preorder:ECBADFHG
4. Inorder: QBKCFAGPEDHR Preorder:GBQACKFPDERH
5. Inorder:DFEGBCAIHKJL Preorder :ABDEFGCHIJKL
45
6. Inorder: A*B-C+D*E/F Preorder: +*A-BC/*DEF
7. Inorder: A*B/C$D*E-F$G+H Preorder: $/*ABC-*DE+$FGH
8. Inorder: A$C*D/E+F-G*H
Preorder: /$A*CD+E-F*GH
46
Create Binary Tree1. Inorder: BAC Postorder:BCA
2. Inorder: ABCDFEG Postorder: ACBFGED
3. Inorder: BACDEHFG Postorder: ABDCHGFE
4. Inorder: BCADFEHGI Postorder:CABFHIGED
5. Inorder: BACDEFGIHKJL
Postorder: ABDCFEIKLJHG
47
6. Inorder: A+B-C*D$E*F Postorder:ABC-DEF*$*+
7. Inorder: A+B-C$D+E-F*G$H Postorder: ABC-D$*EFG*H$-+
8. Inorder: A*B-C$D/E+F
Postorder: ABC-*DEF+/$
BINARY SEARCH TREES
48
15/1/201416/1/2014
49
Binary Search Tree(BST)Binary search tree(BST) is a binary tree that is empty or each node satisfies the following properties:
1) every element has a key, and no two elements have the same key
2) the keys in a nonempty left subtree must be smaller than the key in the root of the subtree
3) the keys in a nonempty right subtree must be larger than the key in the root of the subtree
4) the left and right subtrees are also BST
50
E.g.
1. 14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
2. 18, 5, 9, 19, 4, 3, 30, 16, 7, 26, 4, 25, 2, 35
3. 10, 5, 8, 25, 4, 18, 13, 9, 1, 3, 12
4. 80, 90, 45, 60, 115, 78, 92, 69, 70, 35, 22, 145, 96, 55, 82
51
Operations on BSTInit
Create
Find/Search
Makeempty
Insert
Delete
Findmin
Findmax
52
BST as an ADTclass BST
{
struct BSearchT
{ char Info[10]; BSearchT *Left,*Right;
BSearchT(){ Left=Right=NULL; }
} *Root;
public:
BST(){Root=NULL;}//init
void Makeempty(); // make tree empty
void Create();
void Insert(char*); //Binary Search Tree Creation
int Search(char *);//Non Recursive Search
int Rsearch(char *); //Recursive Search
53
void DFS();//Depth First Search Non Recursive
void RDFS();//Depth First Search Recursive
void BFS();//Breath First Search Non Recursive(Levelwise Display)
//Count Number of Node,No of Leaf Node & Height of the Tree
void BST::RBFS();//Breath First Search Recursive
void Mir_RBFS(); //Display Mirror image of Tree Recursively
void mirror(); //Display Mirror image of Tree Non Recursive
void mir_BFS(); //Non Recursive Mirror Image
void R_mir() //Mirror Image Recursive
BSearchT * Dsearch(char *M) //Delete Node
BSearchT *Tree_copy() //Tree Copy
};
54
Makeempty() void BST::Makeempty()
{
If(Root!=NULL)
{
Root=Root->Left;
Makeempty();
Root=Root->Right;
Makeempty();
Delete Root;
}
}
55
Create //create tree
void BST::Create()
{
char Str[10],ans;
do
{ cout<<"Enter String";
cin>>Str;
Insert(Str);
cout<<"\nAdd More...(Y/N)";
cin>>ans;
}while(ans=='y' || ans=='Y');
}
56
insert//Binary Search Tree Creation
void BST::Insert(char* String)
{
BSearchT *Node,*Temp;
Node=new BSearchT; strcpy(Node->Info,String);
if (Root==NULL) Root=Node;
else { Temp=Root;
while(1)
{
if(strcmp(Node->Info,Temp->Info)>0)
{ if(Temp->Right==NULL)
{Temp->Right=Node; break;}
else Temp=Temp->Right;
}
57
Else if(strcmp(Node->Info,Temp->Info)<0)
{
if(Temp->Left==NULL)
{ Temp->Left=Node; break;}
else Temp=Temp->Left;
}
}//while
}
}
58
Search //Non Recursive Search
int BST::Search(char *M)
{
int Flag=0; BSearchT *Temp=Root;
while(Temp!=NULL)
{
if(strcmp(M,Temp->Info)>0)
Temp=Temp->Right;
else if(strcmp(M,Temp->Info)<0)
Temp=Temp->Left;
else { Flag=1;break;}
}
return Flag;
}
59
//Recursive Search
int BST::Rsearch(char *M)
{
static int Flag=0;
static BSearchT *Temp=Root;
if(Temp!=NULL)
{
if(strcmp(M,Temp->Info)>0)
{ Temp=Temp->Right;Rsearch(M);}
else if(strcmp(M,Temp->Info)<0)
{ Temp=Temp->Left; Rsearch(M);}
else Flag=1;
}
return Flag;
}
60
DFS//Depth First Search Non Recursive
void BST::DFS()
{
BSearchT *Stack[20],*Temp=Root; int Top=-1;
while(Temp!=NULL || Top > -1)
{
cout<<Temp->Info;
if(Temp->Right!=NULL)
Stack[++Top]=Temp->Right;
Temp=Temp->Left;
if(Temp==NULL)
Temp=Stack[--Top];
}
}
18-1-2014
61
//Depth First Search Recursive
void BST::RDFS()
{
static BSearchT *Temp=Root;
if(Temp!=NULL)
{
cout<<Temp->Info;
Temp=Temp->Left;
RDFS();
Temp=Temp->Right;
RDFS();
}
}
62
BFS//Breath First Search Non Recursive(Level wise Display)
//Count Number of Node, No of Leaf Node & Height of the Tree
void BST::BFS()
{
BSearchT *Queue[20],*Temp=Root;
int Front=0,Rear=1,Level=-1,Count=0,Leaf=0;
if(Temp==NULL)
cout<<"\nEmpty Temp";
else
{
Queue[0]=Temp;
Queue[1]=NULL;
cout<<"\nLevel "<<++Level;
63
do
{ Temp=Queue[Front++];
if(Temp==NULL)
{ Queue[++Rear]=NULL; cout<<"\nLevel "<<++Level; }
else
{ Count++;//Total Number of Nodes in a Tree
cout<<" "<<Temp->Info<<" ";
if(Temp->Left!=NULL) Queue[++Rear]=Temp->Left;
if(Temp->Right!=NULL) Queue[++Rear]=Temp->Right;
else if(Temp->Right==NULL && Temp->Left==NULL)
Leaf++;//Number of Leaf Nodes
}
}while(Front<Rear);
}
64
cout<<"\nHeight of Tree "<<Level-1;
cout<<“\nTotal No of Nodes"<<Count;
cout<<"\n Number of leaf Nodes "<<Leaf;
}
65
//Breath First Search Recursive
void BST::RBFS()
{
static int Front=0,Rear=0;
static BSearchT *Queue[10],*Temp=Root;
if(Front<=Rear)
{ cout<<" "<<Temp->Info<<" ";
if(Temp->Left!=NULL) Queue[Rear++]=Temp->Left;
if(Temp->Right!=NULL) Queue[Rear++]=Temp->Right;
Temp=Queue[Front++];
RBFS();
}
}
66
Display Mirror//Display Mirror image of Tree Non Recursive
void BST::mirror()
{
struct BSearchT *Queue[20],*Temp=Root;
int Front=0,Rear=1;
Queue[0]=Temp; Queue[1]= NULL;
while(Front<Rear)
{
Temp=Queue[Front++];
if(Temp==NULL)
{ cout<<"\n";
Queue[++Rear]=NULL;
67
}
else
{
cout<<" "<<Temp->Info<<" ";
if(Temp->Right!=NULL)
Queue[++Rear]=Temp->Right;
if(Temp->Left!=NULL)
Queue[++Rear]=Temp->Left;
}
}
}
68
Change to Mirror//Non Recursive Mirror Image
void BST::mir_BFS()
{
struct BSearchT *Queue[20],*Temp1,*Temp=Root; int Front=0,Rear=1;
Queue[0]=Temp;
do
{
Temp=Queue[Front++];
Temp1=Temp->Left;
Temp->Left=Temp->Right;
Temp->Right=Temp1;
if(Temp->Left!=NULL) Queue[Rear++]=Temp->Left;
if(Temp->Right!=NULL) Queue[Rear++]=Temp->Right;
}while(Front<Rear);
}
69
Delete
Leaf Node (Left & Right NULL)
One Child Node(Left or Right NULL)
Two Children Node(Left & Right not NULL)
70
Binary Search Tree(BST) Deleting a leaf or a node with 1 child
30
5
2
40
80
30
5
2
80
30
5
2
40
8035
(a) delete 35 (leaf) (b) delete 40 (node with single child)
71
Binary Search Tree(BST)
deletion of a node with two children
40
20
10
60
30 50 70
45 55
52
40
20
10
55
30 50 70
45 52
(a) tree before deletion of 60 (b) tree after deletion of 60
Threaded Binary Tree
73
TBTThere are more null links than actual pointers in representation of any binary tree
n+1 null links out of 2n total links
How to use the null links?
Replace the null links by pointers, called threads, to other nodes in the tree
74
TBT
8
75
3
11
13
1
6
9
75
Aroot
B C
GE
I
D
H
Fdangling
dangling
inorder traversal:H, D, I, B, E, A, F, C, G
TBT
76
1 1--
1 1A
1 1C1 1B
0 0E 0 0F 0 0G1 1D
0 0I0 0H
head
Memory Representation of TBT
root
77
Applications/Uses of treesTrees are used in computer programming.
They can be used for improving database search times. (e.g. binary search trees, AVL trees, red-black trees).
Game programming(minmax trees, decision trees, path finding trees)
Graphics programming (BSP trees, Quadtrees, octrees)
Arithmatic scripting languages (arithmatic precedence trees)
Data compression (Huffman trees)
File systems (B-trees, sparse indexed trees, trie trees)
78
Thank You
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