1. Data Structures: A Pseudocode Approach with C, Second Edition1 Objectives Upon completion you will be able to: Explain the differences between a BST and an AVL tree Insert, delete, and process nodes in an AVL tree Understand the operation of the AVL tree ADT AVL Search Trees

2. Data Structures: A Pseudocode Approach with C, Second Edition2 AVL Tree Basic Concepts While BST is simple and easy to understand, it has one major problem which is not balanced. In 1968, two Russian mathematicians, G.M. Adelson-Velskii and E.M. Landis, created the balanced binary tree structure that is AVL tree. An AVL tree is a BST that is guaranteed to always be balanced. An AVL tree is a binary tree that either is 1. empty or 2. consists of two AVL subtrees, T L and T R, whose heights differ by no more than 1 | H L – H R | ≤ 1

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4. 4 8-1 AVL Tree Basic Concepts AVL Tree Balance Factor Balancing Trees

5. Data Structures: A Pseudocode Approach with C, Second Edition5 AVL Tree Balance Factor The balance factor = H L – H R The balance factor for any node in AVL tree must be +1, 0, -1 The descriptive identifiers are as follows: LH for left high (+1) indicates the left subtree is higher than the right subtree. EH for even high (0) indicates the subtrees are the same height. RH for right high (-1) indicates the left subtree is shorter than the right subtree.

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7. 7 Balancing Trees AVL trees are balanced by rotating nodes either to the left or to the right Four cases that require rebalancing. All unbalanced trees fall into one of these four cases: 1. Left of left: A subtree of a tree that is left high has also become left high.Left of left 2. Right of right: A subtree of a tree that is right high has also become right high.Right of right 3. Right of left: A subtree of a tree that is left high become right high.Right of left 4. Left of right: A subtree of a tree that is right high become left high.Left of right

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9. 9 (continued)

10. Data Structures: A Pseudocode Approach with C, Second Edition10 Case 1: Left of Left

11. Data Structures: A Pseudocode Approach with C, Second Edition11 Case 2: Right of Right

12. Data Structures: A Pseudocode Approach with C, Second Edition12 Case 3: Right of Left

13. Data Structures: A Pseudocode Approach with C, Second Edition13 Case 4: Left of Right

14. Data Structures: A Pseudocode Approach with C, Second Edition14 AVL Tree Implementations Insertion and deletion in an AVL tree follow the same basic rules for insertion and deletion in a BST. The difference lies in the tree balancing, which occurs as we back out of the tree. In this section we develop the insertion and deletion algorithms for a AVL tree, including algorithms to balance the tree. Insert into AVL Tree AVL Tree Delete Algorithm

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