Avl handout

Apr 10, 20131 like306 views

The document discusses AVL trees and provides examples of inserting nodes into an AVL tree. It shows how single and double rotations are used to rebalance the tree when inserts cause height imbalances. Single rotations involve rotating a node and its child, while double rotations rotate a node and its child and grandchild.

CS216 Fall 2001

AVL Trees Example

Insert 3 Insert 2 Insert 1 (non-AVL) AVL

3 3 3 2

2 2 Single 1 3
rotation
1

Insert 4 Insert 5 (non-AVL) AVL

2
2 2

Single 1 4
3 1 3
1 rotation

3 5
4 4

5

Insert 6 (non-AVL) AVL Insert 7 (non-AVL)

4 4
2

Single 2 5 2 5
1 4 rotation

1 3 6 1 3 6
3 5

7
6

9/27/01 Page 1 of 4

CS216 Fall 2001

AVL Insert 16 Insert 15 (non-AVL)

4
4 4
Single
rotation 2 6
2 6 2 6

1 3 5 7
1 3 5 7 1 3 5 7

16 16

15

Step 1: Rotate child and grandchild Step 2: Rotate node and new child (AVL)

4 4

Double 2 6 2 6
rotation

1 3 5 7 1 3 5 15

15 7 16

16

Insert 14 (non-AVL) Step 1: Rotate child and Step 2: Rotate node and
grandchild new child (AVL)

4
4 4
Double
rotation 2 7
2 6 2 6

1 3 6 15
1 3 5 15 1 3 5 7

16
7 16 15 5 14

14 14 16

9/27/01 Page 2 of 4

CS216 Fall 2001

AVL Tree No longer AVL, must rebalance

Insert a node

h h +1 h h +2

Two ways to expand subtree of height h+2:

h h

h h +1 h +1 h
h +2
h +2
Apply a Single Rotation Apply a Double Rotation

Note: the symmetrical case is handled
identically (i.e. mirror image)

9/27/01 Page 3 of 4

CS216 Fall 2001

Single Rotation:

B D

Single
rotation
D B

A
E

h h +1
C E A C

h h +1 h h

h +2
Double Rotation:

B
D

F
B F
A Double
rotation
D
h

G A C E G

h h h h -1 h
C E OR
h-1 h
h h -1 OR
h h
OR
h-1 h
OR
h h
h +1

h +2
9/27/01 Page 4 of 4

- AVL trees are binary search trees where the heights of the two subtrees of every node differ by at most one. This balances the tree and ensures search, insertion, and deletion operations take O(log N) time. - Insertions can cause imbalance, requiring rotations to restore balance. The node nearest to the inserted node that becomes imbalanced is rotated. Single and double rotations are used depending on insertion location. - The number of nodes in an AVL tree of height h is approximately equal to the hth Fibonacci number. Since the Fibonacci numbers grow exponentially with h, h must be O(log N) to contain N nodes.

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