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CS223 Advanced Data Structures and Algorithms

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1 CS223 Advanced Data Structures and Algorithms
AVL Tree Neil Tang 02/03/2010 CS223 Advanced Data Structures and Algorithms

2 CS223 Advanced Data Structures and Algorithms
Class Overview Definition Tree height Tree rotation: single and double Insertion with rotations CS223 Advanced Data Structures and Algorithms

3 CS223 Advanced Data Structures and Algorithms
Definition An AVL tree is a special binary search tree in which for each node, the heights of its left and right subtree can differ by at most 1. CS223 Advanced Data Structures and Algorithms

4 CS223 Advanced Data Structures and Algorithms
Definition CS223 Advanced Data Structures and Algorithms

5 CS223 Advanced Data Structures and Algorithms
Height Upper bound: 1.44log(N+2)-1.328 In practice, slightly more than logN. The complexity of an AVL tree operation is usually O(height). CS223 Advanced Data Structures and Algorithms

6 Cases Causing Violation
Case 1 (left-left): An insertion to the left subtree of the left child of a node. Case 2 (left-right): An insertion to the right subtree of the left child of a node. Case 3 (right-left): An insertion to the left subtree of the right child of a node. Case 4 (right-right): An insertion to the right subtree of the right child of a node. CS223 Advanced Data Structures and Algorithms

7 Single Rotation To Fix Case 1
CS223 Advanced Data Structures and Algorithms

8 Single Rotation To Fix Case 1
CS223 Advanced Data Structures and Algorithms

9 Single Rotation To Fix Case 4
CS223 Advanced Data Structures and Algorithms

10 Single Rotation To Fix Case 4
CS223 Advanced Data Structures and Algorithms

11 Left-Right Double Rotation To Fix Case 2
CS223 Advanced Data Structures and Algorithms

12 Right-Left Double Rotation To Fix Case 3
CS223 Advanced Data Structures and Algorithms

13 Right-Left Double Rotation To Fix Case 3
CS223 Advanced Data Structures and Algorithms

14 CS223 Advanced Data Structures and Algorithms
Right Single Rotation CS223 Advanced Data Structures and Algorithms

15 Left-Right Double Rotation
CS223 Advanced Data Structures and Algorithms

16 Insertion with Rotations
CS223 Advanced Data Structures and Algorithms

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