Computer notes - AVL Tree

Class No.20 Data Structures http://ecomputernotes.com

AVL Tree
AVL (Adelson-Velskii and Landis) tree.
An AVL tree is identical to a BST except
height of the left and right subtrees can differ by at most 1.
height of an empty tree is defined to be (–1).
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AVL Tree
An AVL Tree
height 0 1 2 3 http://ecomputernotes.com 5 8 2 4 3 1 7

AVL Tree
Not an AVL tree
height 0 1 2 3 http://ecomputernotes.com 6 8 1 4 3 1 5

Balanced Binary Tree
The height of a binary tree is the maximum level of its leaves (also called the depth).
The balance of a node in a binary tree is defined as the height of its left subtree minus height of its right subtree.
Here, for example, is a balanced tree. Each node has an indicated balance of 1, 0, or –1.

Balanced Binary Tree http://ecomputernotes.com -1 0 1 0 0 0 0 -1 0 1 0 0 0 0 0 0 0

Insertions and effect on balance
U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 U 9 U 10 U 11 U 12 B B B B B B http://ecomputernotes.com -1 0 1 0 0 0 0 -1 0 1 0 0 0 0 0 0 0

Tree becomes unbalanced only if the newly inserted node
is a left descendant of a node that previously had a balance of 1 (U 1 to U 8 ),
or is a descendant of a node that previously had a balance of –1 (U 9 to U 12 )

Insertions and effect on balance

Consider the case of node that was previously 1

Inserting New Node in AVL Tree A B T 3 1 http://ecomputernotes.com 1 0 T 1 T 2

Inserting New Node in AVL Tree A B T 3 new 1 2 http://ecomputernotes.com 2 1 T 1 T 2

Inserting New Node in AVL Tree A B T 3 new 1 2 A B T 3 new Inorder: T 1 B T 2 A T 3 Inorder: T 1 B T 2 A T 3 http://ecomputernotes.com 2 1 T 1 T 2 0 0 T 1 T 2

AVL Tree Building Example
Let us work through an example that inserts numbers in a balanced search tree.
We will check the balance after each insert and rebalance if necessary using rotations.

Insert(1)
http://ecomputernotes.com 1

Insert(2)
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Insert(3) single left rotation
-2 http://ecomputernotes.com 1 2 3

Insert(3)
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Computer notes - AVL Tree

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