C-Programming-Class 14

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Session 14

Trees


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Session Objectives

To understand the concepts of trees.

To know different parts of trees.


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Session Topics

Introduction to trees

Parts of trees

Binary trees

In-order Transversal

Post-order Transversal Threaded Binary Tree


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Definition of a Tree

A directed tree is an acyclic graph, which has only one

node with indegree 0 (root) and all other nodes haveindegree 1.

Indegree the number of edges incidenton a node .Outdegree the number of edges leavinga node.


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Parts of a Tree


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nodes

Parts of a Tree


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parent

node

Parts of a Tree


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child

nodesparent

node

Parts of a Tree


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child

nodesparent

node

Parts of a Tree


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root node

Parts of a Tree


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leaf

nodes

A node that has an outdegree of 0, is

called a Leaf node or Terminal node.

Parts of a Tree


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sub-tree

Parts of a Tree


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sub-tree

Parts of a Tree


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sub-tree

Parts of a Tree


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10

20 15

2530

Level It is number of edges in the path from the root node

Ex: Level of 25 is 2, Level of 10 is 0

Height It is one more than the maximum level in the tree.

Ex: Height of tree shown below is 3.

Level & Height of a Tree


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Binary Tree

A tree in which outdegree of each node is less than or equal to

two is called a Binary tree.

If the outdegree of every node is either 0 or 2, it is called Strictly

Binary Tree.


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Storage representation of a node

In a linked allocation technique, a node in a tree has threefields.

infowhich contains the actual information

llinkwhich contains address of the left subtree

rlinkwhich contains address of the right subtree

A node can be represented as a structure as follows

struct node{

int info;struct node *llink;struct node *rlink;

};


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Various operations performed on Trees

Insertion Insert a given item into a tree

Traversal Visiting the nodes of the tree

one by one Search Search for the specified item

CopyingTo obtain exact copy of the

given tree

Deletion To delete a node from the tree


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Traversals

Traversal is the most common operation that can be

performed on trees.

There are three types of traversals:-

Preorder

Inorder

Postorder

Preorder Traversal

Recursive definition of Preorder traversal

Process the root Node[N]

Traverse the Left subtree in preorder[L]

Traverse the Right subtree in preorder[R]


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Example: Preorder

FIECHGDBA

A

B C

D E

HG

F

I


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Function for preorder traversal

void preorder(NODE root)

{

if(root != NULL)

{

printf(%d,root->info);

preorder(root->llink);

preorder(root->rlink);}

}


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Inorder Traversal

Traverse the Left subtree in inorder[L]


Traverse the Right subtree in inorder[R]


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Example: Inorder

FCIEABHDG

A

B C

D E

HG

F

I


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Function for inorder traversal

void inorder(NODE root)

{

if(root != NULL){

inorder(root->llink);

printf(%d,root->info);

inorder(root->rlink);}

}


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Postorder Traversal

Traverse the Left subtree in postorder[L]

Traverse the Right subtree in postorder[R]



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ACFEIBDHG

Example:Postorder

A

B C

D E

HG

F

I


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Insertion

Algorithm to insert an item into a tree

Create new node for the item.

Find a parent node according to thedirectionsgiven.

Find the appropriate position to insert the node

Attach the new node as a leaf.


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31 64

20 40 56

28 33 47 59

89

57Example:

Insert


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Insert

Example: 5743

31 64

20 40 56

28 33 47 59

89

57

43

64

56

59

57


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Binary Search Tree

A binary search tree is a binary tree in which foreach node x in the tree, elements in the left subtreeare less than info(x) and elements in the rightsubtree are greater than or equal to info(x).

Various operations performed

Insertion

Searching

Deleting


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35 64

25 40 56

28 32 47 59

89

Binary Search Tree


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Insertion

25

140

100

50

90

80

200

300150

180140


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Function to insert an itemNODE insert(int item, NODE root){

NODE temp,cur,prev;temp = get_node(); /*Obtain a new node*/

temp->info = item;

temp->llink = NULL;

temp->rlink = NULL;

if(root == NULL)

return temp;

prev = NULL; /*find the appropriate position*/cur = root;

while(cur != NULL)

{

prev = cur;/*Obtain parent and left or right child*/

cur = (item < cur->info) ? Cur->llink :cur->rlink;

}

if(item < prev->info) /*new node is less than parent */prev->llink = temp; /*Insert towards left */

else

prev->rlink = temp; /*Insert towards right */

return root; /*Return the root*/

}


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Deletion

When we delete a node, we need to consider how we

take care of the children of the deleted node.

This has to be done such that the property of the

search tree is maintained.


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Deletion

Two cases:

(1) the node is a leaf

Delete it immediately

(2) the node has one child

Adjust a pointer from the parent to bypass that node


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Threaded Binary Tree

Inorder traversal of a threaded binary tree By using threads we can perform an inorder traversal

without making use of a stack (simplifying the task)

Now, we can follow the thread of any node,ptr, to thenext node of inorder traversal

Ifptr->right_thread= TRUE, the inorder successor ofptrisptr->right_childby definition of the threads

Otherwise we obtain the inorder successor of ptr byfollowing a path of left-child links from the right-child ofptruntil we reach a node with left_thread= TRUE


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Summary

A directed tree is an acyclic graph, which has onlyone node with indegree 0 (root) and all othernodes have indegree 1.

A node that has an outdegree of 0, is called a Leafnode or Terminal node.

A tree in which outdegree of each node is less thanor equal to two is called a Binary tree.

If the outdegree of every node is either 0 or 2, it iscalled Strictly Binary Tree.


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Thank You!

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C-Programming-Class 14