AVL Tree Explained
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An AVL tree is a self-balancing binary search tree that maintains a height difference of no more than one between its left and right subtrees to ensure efficient operations. Various types of tree rotations (left, right, left-right, right-left) are employed to maintain this balance when nodes are added. The document outlines the insertion process and illustrates examples of AVL tree construction and balancing through rotations.
AVL Tree Explained
- 1. AVL TREE Prof. Manjusha Amritkar Information Technology Department International Institute of Information Technology, I²IT www.isquareit.edu.in
- 2. Binary Search Tree • E.g. 2,4,6,8,10 • Right skewed tree • Time needed to perform insertion ,deletion and many other operations is O(N) in worst case. • We need tree with small height with Log N • Such trees are called as Balanced Binary Search Tree • e.g AVL tree, Red Black Tree International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected] 2 4 6 8 10
- 3. AVL Tree • AVL tree Named after their inventor Adelson, Velski & Landis, is a self-balancing Binary Search Tree (BST) where the difference between heights of left and right subtrees cannot be more than one for all nodes. • The difference between the heights is called as balanced factor (BF) • BF = height (left subtree) – height (right subtree) • The tree is balanced when BF is -1, 0, 1 International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected]
- 4. AVL Tree Rotations • If the difference is more than one then the tree is to be balanced using rotations. • Left rotation (LL) Right rotation (RR) Left-Right rotation (LR) Right-Left rotation (RL) International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected]
- 5. Left Rotation • If a tree becomes unbalanced, when a node is inserted into the right subtree of the right subtree, then we perform a single left rotation International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected]
- 6. Right Rotation • AVL tree may become unbalanced, if a node is inserted in the left subtree of the left subtree. The tree then needs a right rotation International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected]
- 7. Left Right Rotations • A node has been inserted into the right subtree of the left subtree. This makes C an unbalanced node. These scenarios cause AVL tree to perform left-right rotation. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected] LR Rotation
- 8. Right Left Rotations • A node has been inserted into the left subtree of the right subtree. This makes A, an unbalanced node with balance factor 2. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected] RL Rotation
- 9. Example: Construct an AVL Tree by inserting numbers from 1 to 8. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected]
- 10. Example: Construct an AVL Tree by inserting numbers from 1 to 8. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected]
- 11. Example: Construct an AVL Tree by inserting numbers from 1 to 8. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected]
- 12. Example: Construct an AVL Tree by inserting numbers from 1 to 8. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected]
- 13. Example: Construct an AVL Tree by inserting numbers from 1 to 8. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected]
- 14. Example: Construct an AVL Tree by inserting numbers from 1 to 8. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected]
- 15. Example: Construct an AVL Tree by inserting numbers from 50,25,10,5,7,3,30,20,8,15. • Insert 50 • Insert 25 50 0 Tree is balanced 50 1 Tree is balanced 25 0 International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected]
- 16. Example: Construct an AVL Tree by inserting numbers from 50,25,10,5,7,3,30,20,8,15. • Insert 10 Tree is imbalanced Tree is balanced International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected] 50 2 25 1 10 0 25 1 10 0 50 0 RR Rotation
- 17. Example: Construct an AVL Tree by inserting numbers from 50,25,10,5,7,3,30,20,8,15. • Insert 5 insert 7 Tree is balanced Tree is imbalanced 25 1 10 1 50 0 5 0 25 1 7 0 50 0 5 0 10 0 Tree is balanced International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected] 25 2 10 2 50 0 5 -1 7 0 LR Rotation
- 18. Example: Construct an AVL Tree by inserting numbers from 50,25,10,5,7,3,30,20,8,15. Tree is imbalanced 25 2 7 1 50 0 5 1 7 0 5 25 0 3 0 0 Tree is balanced 10 0 3 0 50 0 1 10 International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected] • Insert 3 LL Rotation
- 19. Example: Construct an AVL Tree by inserting numbers from 50,25,10,5,7,3,30,20,8,15. • Insert 30 insert 20 Tree is imbalanced Tree is balanced 7 -1 5 25 -1 3 0 0 50 1 1 10 0 30 20 0 10 0 7 25 0 5 1 1 50 0 1 0 30 50 0 8 15 0 0 3 0 International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected] 7 -2 5 25 1 3 0 -1 50 1 1 10 30 1 20 15 8 RL Rotation
- 20. References • http://www.btechsmartclass.com/data_structures/avl-trees.html International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected]
- 21. International Institute of Information Technology, I²IT, P-14, Rajiv Gandhi Infotech Park, Hinjawadi Phase 1, Pune - 411 057 Phone - +91 20 22933441/2/3 | Website - www.isquareit.edu.in | Email - [email protected] E-mail: [email protected] Website: http://www.isquareit.edu.in/ Thank You