data structures 3

CS1301 Data Structures

Unit III Non LINEAR DATA STRUCTURE

Preliminaries Binary Tree The Search Tree ADT Binary Search Tree AVL Tree Tree Traversals Hashing - General Idea Hash Function Separate Chaining OpenAddressing Linear Probing Priority Queue (Heaps) - Model - Simple Implementations Binary Heap

Trees• Finite set of one or more nodes that there is a

specially designated node called root and zero or more non empty sub trees T1, T2,.. each of whose roots are connected by a directed edge from Root R

• A tree T is a set of nodes storing elements such that the nodes have a parent-child relationship that satisfies the following– if T is not empty, T has a special tree called the root

that has no parent– each node v of T different than the root has a unique

parent node w; each node with parent w is a child of w

Trees• Root : node which doesn’t have a parent• Node : Item of information• Leaf/ terminal : node which doesn’t have children • Siblings : children of same parents • Path : sequence of nodes n1,n2,n3,….,nk such

that ni is parent of ni+1 for 1<= i<k. there is exactly only one path from each node to root; path from A to L is A,C,F,L

• Length : no. of edges on the path ; length of A to L is 3

• Degree : number of subtrees of a node; A -> 4, C->2 degree of tree is maximum no.of any node in the tree ; degree of tree is 4

Trees• Level : initially letting the root be at level one,

if a node is at level L then its children are at level L + 1– Level of A is 1; Level of B, C, D is 2; Level of

F,G,H,I,J is 3, Level of K,L,M is 4

• Depth : For any node n, the depth of n is the unique path from root to n– Depth of root is 0; Depth of L is 3

• Height : For any node n, the height of the node is the length of the longest path from n to the leaf– Height of leaf is 0, height of node F is 1; height of L

is 0

CHAPTER 5 7

Level and Depth

K L

E F

B

G

C

M

H I J

D

A

Level

1

2

3

4

node (13)degree of a nodeleaf (terminal)nonterminalparentchildrensiblingdegree of a tree (3)ancestorlevel of a node

3

2 1 3

2 0 0 1 0 0

0 0 0

1

2 2 2

3 3 3 3 3 3

4 4 4

Trees

• Binary Tree–Tree in which no node can have more than two children

Trees

• Binary Tree Node Declaration

Struct Treenode{ int element;

Struct Treenode *Left;Struct TreeNode *Right;

};

Trees

• Binary Tree–Comparison between general tree & Binary Tree

General Tree Binary Tree

Has any no. of children Has not more than two children

Trees

• Full Binary Tree– Full binary tree of h has 2^h+1 -1 nodes

Trees• Complete Binary Tree– A complete binary tree is a binary tree,

which is completely filled, with the possible exception of the bottom level, which is filled from left to right

– A full binary tree can be a complete binary tree, but all complete binary tree is not a full binary tree

Trees• Representation of Binary Tree–Two ways• Linear Representation • Linked Representation

Trees• Linear Representation of

Binary Tree– Elements are represented using arrays– For any element in position i, the left

child is in position 2i, right child is in position (2i+1) and parent in position (i/2)

Trees• Linked Representation of Binary Tree– Elements are represented using pointers– Each node in linked representation has two fields

• Pointer to left subtree• Data field• Pointer to right subtree

– In leaf node, both pointer fields are assigned as NULL

Trees• Expression Tree– Is a binary tree– Leaf nodes are operands and interior

nodes are operators– Expression tree be traversed by

inorder , preorder, postorder traversal

Trees• Constructing an Expression Tree

1. Read one symbol at a time from postfix expression

2. Check whether symbol is an operand or operator

1. If operand, create a one –node tree and push a pointer on to the stack

2. If operator, pop two pointers from the stack namely T1 and T2 and form a new tree with root as the operator and T2 as a left child and T1 as a right child. A pointer to this new tree is pushed onto the stack

Trees• Example

Trees• Binary Search Tree– Is a binary tree –For every node x in the tree, value of all the

keys in its left subtree are smaller than the value X and value of all the keys in the right subtree are larger than the value X

Trees• Binary Search Tree–Every binary search tree is a binary tree–All binary tree need not be bnary search tree

Trees• Declaration Routine for Binary

Search Tree

Trees• Routine for Make an empty tree

Trees• Insertion

Trees• Insertion - Example

Trees• Routine for Insertion

Trees• Find

Trees• Find – Example to find 10

Trees• Routine for Find

Trees• Find Minimum–Returns position of the smallest element

in a tree–Start at the root and go left as long as

there is a left child, the stopping point will be the smallest element

Trees• Find Minimum – example

Trees• Find Minimum – Recursive

routine

Trees• Find Minimum – Non

Recursive routine

Trees• Find Maximum–Returns the largest element in the tree–To perform, start at the root and go

right as long as there is a right child–Stopping point is the largest element

Trees• Find Maximum - Example

Trees• Find Maximum – Recursive

Routine

Trees• Find Maximum – Non

Recursive Routine

Trees• Deletion–Complex in BST–To delete an element, consider the following 3 possibilities•Node to be deleted is a leaf node •Node with one child•Node with two children

Trees• Deletion– Node to be deleted is a leaf node

Trees• Deletion

•Node with one child : deleted by adjusting its parent pointer that points to its child node

Trees• Deletion

•Node with two children : to replace the data of node to be deleted with its smallest data of right subtree and recursively delete that node

Trees• Deletion

•Node with two children

Trees• Deletion Routine

Trees• AVL Tree (Adelson - Velskill and Landis)–Binary search tree except that for every

node in the tree, the height of left and right subtrees can differ by atmost 1–Balance factor is the height of left

subtree minus height of right subtree; -1, 0, or +1– If Balance factor is less than or greater

than 1, the tree has to be balanced by making single or double rotations

Trees• AVL Tree (Adelson Velskill and

Landis)

Trees• AVL Tree (Adelson Velskill and Landis)– AVL tree causes imbalance, when anyone of the

following conditions occur• An insertion into the left subtree of the left child

of node• An insertion into the right subtree of the left

child of node• An insertion into the left subtree of the right

child of node• An insertion into the right subtree of the right

child of nodeImbalances can overcome by 1. Single rotation 2. Double rotation

Trees• AVL Tree (Adelson Velskill and

Landis)• Single rotation– Performed to fix case1 and case 4Case 1: An insertion into the left subtree of the left child of node

data structures 3

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