AVL Tree Neil Tang 02032010 CS 223 Advanced

AVL Tree Neil Tang 02/03/2010 CS 223 Advanced Data Structures and Algorithms 1

Class Overview Ø Definition Ø Tree height Ø Tree rotation: single and double Ø Insertion with rotations CS 223 Advanced Data Structures and Algorithms 2

Definition Ø 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. CS 223 Advanced Data Structures and Algorithms 3

Definition CS 223 Advanced Data Structures and Algorithms 4

Height Ø Upper bound: 1. 44 log(N+2)-1. 328 Ø In practice, slightly more than log. N. Ø The complexity of an AVL tree operation is usually O(height). CS 223 Advanced Data Structures and Algorithms 5

Cases Causing Violation Ø Case 1 (left-left): An insertion to the left subtree of the left child of a node. Ø Case 2 (left-right): An insertion to the right subtree of the left child of a node. Ø Case 3 (right-left): An insertion to the left subtree of the right child of a node. Ø Case 4 (right-right): An insertion to the right subtree of the right child of a node. CS 223 Advanced Data Structures and Algorithms 6

Single Rotation To Fix Case 1 CS 223 Advanced Data Structures and Algorithms 7

Single Rotation To Fix Case 1 CS 223 Advanced Data Structures and Algorithms 8

Single Rotation To Fix Case 4 CS 223 Advanced Data Structures and Algorithms 9

Single Rotation To Fix Case 4 CS 223 Advanced Data Structures and Algorithms 10

Left-Right Double Rotation To Fix Case 2 CS 223 Advanced Data Structures and Algorithms 11

Right-Left Double Rotation To Fix Case 3 CS 223 Advanced Data Structures and Algorithms 12

Right-Left Double Rotation To Fix Case 3 CS 223 Advanced Data Structures and Algorithms 13

Right Single Rotation CS 223 Advanced Data Structures and Algorithms 14

Left-Right Double Rotation CS 223 Advanced Data Structures and Algorithms 15

Insertion with Rotations CS 223 Advanced Data Structures and Algorithms 16