CS216 Fall 2001

9/27/01 Page 1 of 4

AVL Trees Example

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

Insert 4 Insert 5 (non-AVL) AVL

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

3 3

2

3

2

1

2

1 3Singlerotation

2

1 3

4

5

1 3

4

Singlerotation

2

1 4

53

2

1 4

53

6

2

Singlerotation

4

2 5

1 3 6

4

2 5

1 3 6

7

CS216 Fall 2001

9/27/01 Page 2 of 4

AVL Insert 16 Insert 15 (non-AVL)

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

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

4

2 6

1 3 75

Singlerotation

16

4

2 6

1 3 75

4

2

1 3 75

16

15

Doublerotation

16

6

4

2

1 3 75

15

6

4

2

1 3 155

167

Doublerotation

4

2

1 3 75

15

14 16

5

4

2

1 3 156

1614

6

6

4

2

1 3 155

167

14

67

CS216 Fall 2001

9/27/01 Page 3 of 4

AVL Tree No longer AVL, must rebalance

Two ways to expand subtree of height h+2:

h h +2

Insert a node

h h +1

h

h h +1

h

h +1 h

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

Apply a Single Rotation Apply a Double Rotation

h +2 h +2

CS216 Fall 2001

9/27/01 Page 4 of 4

Single Rotation:

Double Rotation:

h

h h +1

A

B

D

C E

Singlerotation

E

D

h +1

h h

B

A C

G

F

h

h h -1

D

C E

h

ORh-1 h

ORh h

B

A

D

h h

B

A C

Doublerotation

h -1 h

F

E G

ORh-1 h

OR h h

h +2

h +2

h +1