AVL Tree By Rajanikanth B.

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AVL Trees binary tree for every node x, define its balance factor

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Presentation transcript:

AVL Tree By Rajanikanth B

Introduction AVL Tree is also a Binary Search Tree but it is balanced tree. Here balanced means the difference between height of right subtree and left subtree must be -1 OR 0 OR +1. In AVL Tree every node will have one extra information known as BALANCE FACTOR. BalanceFactor = HeightOfRightSubtree – HeightOfLeftSubtree

The AVL tree is named after its two inventors, G. M The AVL tree is named after its two inventors, G.M. Adelson-Velsky and E.M. Landis, who published it in their 1962. Example

After every insertion / deletion we need to check the BALANCE FACTOR If it is other than -1 or 0 or +1 then we perform ROTATION to make the Tree Balance. Note All AVL Trees are Binary Search Trees but All Binary Search Trees need not be AVL Tree All Binary Search Trees are Binary Trees but All Binary Trees need not be Binary Search Tree

Left Right Rotation (LR) Right Left Rotation (RL) Rotations Left Rotation (LL) Single Rotation Right Rotation (RR) Left Right Rotation (LR) Double Rotation Right Left Rotation (RL)

LL Rotation Insert: A , B , C 2 1 A 1 B C

RR Rotation Insert: C , B , A 2 1 C 1 B A

LR Rotation Insert: C , A , B 2 1 C 1 A B

RL Rotation Insert: A , C , B 2 1 A 1 C B

Construct AVL Tree by inserting 1 to 10 Example Construct AVL Tree by inserting 1 to 10 1 2 3

Deletion Deletion in AVL Tree have 3 cases 1. Deleting node without any child 2. Deleting node with one child 3. Deleting node with two children

1. Deleting node without any child Step – 1: Find given node in AVL Tree by performing search operation Step – 2: Remove given node in AVL Tree by using ‘delete’ operator Step – 3: Check balance factor of each node in AVL tree Step – 4: If tree is not balanced perform suitable Rotation operation to make tree balanced

Deletion Delete: 5 4 2 8 6 9 1 3 5 7 10

2. Deleting node with one child Step – 1: Find given node in AVL Tree by performing search operation Step – 2: Make a link between its Parent and its Child Step – 3: Remove given node in AVL Tree by using ‘delete’ operator Step – 4: Check balance factor of each node in AVL tree Step – 5: If tree is not balanced perform suitable Rotation operation to make tree balanced

Deletion Delete: 6 4 2 8 6 9 1 3 7 10

Deletion Delete: 7 4 2 8 9 1 3 7 10

3. Deleting node with two children Step – 1: Find given node in AVL Tree by performing search operation Step – 2: Find the Smallest node in its Right Subtree Step – 3: Swap both given node and the Smallest node in its right subtree Step – 4: Remove given node in AVL Tree by using ‘delete’ operator Step – 5: Check balance factor of each node in AVL tree Step – 6: If tree is not balanced perform suitable Rotation operation to make tree balanced

Deletion Delete: 4 4 2 9 1 3 8 10

Assignment Construct AVL Tree by inserting 10 to 1 Write applications of AVL Trees Explain Deletion in AVL Tree in detail