Advanced Data Structures - Data Structures - Lecture Slides, Slides of Data Structures and Algorithms

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AVL TREES

By Asami Enomoto

CS 146

AVL Tree is…

• named after A delson- V elskii and L andis
• the first dynamically balanced trees to be propose
• Binary search tree with balance condition in which the sub-trees of each node can differ by at most 1 in their height

An AVL tree has the following properties:

1. Sub-trees of each node can differ by at most 1 in their height
2. Every sub-trees is an AVL tree

AVL tree?

YES Each left sub-tree has height 1 greater than each right sub-tree

NO Left sub-tree has height 3, but right sub-tree has height 1

Single Rotation

left sub-tree is two level deeper than the right sub-tree

move ① up a level and ② down a level

Double Rotation

Left sub-tree is two level deeper than the right sub-tree

Move ② up two levels and ③ down a level

Deletion

• Delete
  • Imbalance at
  • Perform rotation with
• Imbalance at
• Perform rotation with

Key Points

• AVL tree remain balanced by applying rotations, therefore it guarantees O(log N) search time in a dynamic environment
• Tree can be re-balanced in at most O(log N) time