AVL Tree

27th Mar 2007

Outline

• AVL Tree– Definition– Properties– Operations

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AVL Trees• Unbalanced Binary Search Trees are bad. Worst

case: operations take O(n).• AVL (Adelson-Velskii & Landis) trees maintain

balance.• For each node in tree, height of left subtree and

height of right subtree differ by a maximum of 1.

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X X

H

H-1H-2

AVL Trees

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10

5

3

20

2

1 3

10

5

3

20

1

43

5

AVL Trees

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12

8 16

4 10

2 6

14

Insertion for AVL Tree• After insert 1

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12

8 16

4 10

2 6

14

1

Insertion for AVL Tree

• To ensure balance condition for AVL-tree, after insertion of a new node, we back up the path from the inserted node to root and check the balance condition for each node.

• If after insertion, the balance condition does not hold in a certain node, we do one of the following rotations:– Single rotation– Double rotation

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12/26/03 8

Let the node that needs rebalancing be .

There are 4 cases: Outside Cases (require single rotation) : 1. Insertion into left subtree of left child of . 2. Insertion into right subtree of right child of . Inside Cases (require double rotation) : 3. Insertion into right subtree of left child of . 4. Insertion into left subtree of right child of .

The rebalancing is performed through four separate rotation algorithms.

Insertions in AVL Trees

Insertions Causing Imbalance

• An insertion into the subtree:– P (outside) - case 1– Q (inside) - case 2

• An insertion into the subtree:– Q (inside) - case 3– R (outside) - case 4

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RP

Q

k1

k2

Q

k2

P

k1

R

HP=HQ=HR

Single Rotation (case 1)

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A

k2

B

k1

CCBA

k1

k2

HA=HB+1HB=HC

Single Rotation (case 4)

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C

k1

B

k2

AA B C

k2

k1

HA=HB

HC=HB+1

Problem with Single Rotation• Single rotation does not work for case 2 and 3

(inside case)

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Q

k2

P

k1

RR

P

Q

k1

k2

HQ=HP+1HP=HR

Double Rotation: Step

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C

k3

A

k1

D

B

k2

C

k3

A

k1

D

B

k2

HA=HB=HC=HD

Double Rotation: Step

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C

k3

A

k1

DB

k2

Double Rotation

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C

k3

A

k1

D

B

k2

C

k3

A

k1

DB

k2

HA=HB=HC=HD

Double Rotation

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B

k1

D

k3

A

C

k2

B

k1

D

k3

A C

k2

HA=HB=HC=HD

Example• Insert 3 into the AVL tree

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3

11

8 20

4 16 27

8

8

11

4 20

3 16 27

Example• Insert 5 into the AVL tree

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5

11

8 20

4 16 27 8

11

5 20

4 16 27

8

AVL Trees: Exercise

• Insertion order:– 10, 85, 15, 70, 20, 60, 30, 50, 65, 80, 90, 40, 5, 55

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Remove Operation in AVL Tree

• Removing a node from an AVL Tree is the same as removing from a binary search tree. However, it may unbalance the tree.

• Similar to insertion, starting from the removed node we check all the nodes in the path up to the root for the first unbalance node.

• Use the appropriate single or double rotation to balance the tree.

• May need to continue searching for unbalanced nodes all the way to the root.

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Deletion X in AVL Trees

• Deletion:– Case 1: if X is a leaf, delete X– Case 2: if X has 1 child, use it to replace X– Case 3: if X has 2 children, replace X with its

inorder predecessor (and recursively delete it)

• Rebalancing

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Delete 55 (case 1)

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60

20 70

10 40 65 85

5 15 30 50 80 90

55

Delete 55 (case 1)

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60

20 70

10 40 65 85

5 15 30 50 80 90

55

Delete 50 (case 2)

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60

20 70

10 40 65 85

5 15 30 50 80 90

55

Delete 50 (case 2)

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60

20 70

10 40 65 85

5 15 30 50 80 90

55

Delete 60 (case 3)

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60

20 70

10 40 65 85

5 15 30 50 80 90

55

prev

Delete 60 (case 3)

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55

20 70

10 40 65 85

5 15 30 50 80 90

Delete 55 (case 3)

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55

20 70

10 40 65 85

5 15 30 50 80 90

prev

Delete 55 (case 3)

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50

20 70

10 40 65 85

5 15 30 80 90

Delete 50 (case 3)

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50

20 70

10 40 65 85

5 15 30 80 90

prev

Delete 50 (case 3)

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40

20 70

10 30 65 85

5 15 80 90

Delete 40 (case 3)

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40

20 70

10 30 65 85

5 15 80 90

prev

Delete 40 : Rebalancing

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30

20 70

10 65 85

5 15 80 90

Case ?

Delete 40: after rebalancing

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30

7010

20 65 855

15 80 90

Single rotation is preferred!

Minimum Element in AVL Tree• An AVL Tree of height H has at least FH+3-1 nodes,

where Fi is the i-th fibonacci number• S0 = 1• S1 = 2• SH = SH-1 + SH-2 + 1

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SH-1SH-2

H

H-1H-2

AVL Tree: analysis (1)

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328.1)2log(44.1

5/φ

)(

618.12/)5(1φ

5/φ

3H

i

NH

roughlyleastathasHheightofTreeAVL

Fi

AVL Tree: analysis (2)• The depth of AVL Trees is at most logarithmic.• So, all of the operations on AVL trees are also

logarithmic.• The worst-case height is at most 44 percent more

than the minimum possible for binary trees.

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Summary

• Find element, insert element, and remove element operations all have complexity O(log n) for worst case

• Insert operation: top-down insertion and bottom up balancing

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Further Study

• http://telaga.cs.ui.ac.id/WebKuliah/IKI10100/resources/animation/data-structure/avl/avltree.html

• Section 19.4 of Weiss book

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