CS1301 Data Structures - Binary Trees, Binary Search Trees, AVL Trees

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

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 node
leaf (terminal)
nonterminal
parent
children
sibling
degree of a tree (3)
ancestor
level 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
• 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
• Routine for Insertion

Trees
• Find – Example to find 10

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
• 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 node
Imbalances can overcome by
1. Single rotation
2. Double rotation

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

CS1301 Data Structures - Binary Trees, Binary Search Trees, AVL Trees

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CS1301 Data Structures - Binary Trees, Binary Search Trees, AVL Trees