© 2014 Goodrich, Tamassia, Goldwasser AVL Trees 1

AVL Trees

6

3 8

4

v

z

Presentation for use with the textbook Data Structures and Algorithms in Java, 6th edition, by M. T. Goodrich, R. Tamassia, and M. H. Goldwasser, Wiley, 2014

© 2014 Goodrich, Tamassia, Goldwasser AVL Trees 2

AVL Tree Definition

AVL trees are

balanced

An AVL Tree is a

binary search tree

such that for

every internal

node v of T, the

heights of the

children of v can

differ by at most 1

88

44

17 78

32 50

48 62

2

4

1

1

2

3

1

1

An example of an AVL tree where the

heights are shown next to the nodes

© 2014 Goodrich, Tamassia, Goldwasser AVL Trees 3

Height of an AVL TreeFact: The height of an AVL tree storing n keys is O(log n).

Proof (by induction): Let us bound n(h): the minimum number of internal nodes of an AVL tree of height h.

We easily see that n(1) = 1 and n(2) = 2

For n > 2, an AVL tree of height h contains the root node, one AVL subtree of height n-1 and another of height n-2.

That is, n(h) = 1 + n(h-1) + n(h-2)

Knowing n(h-1) > n(h-2), we get n(h) > 2n(h-2). Son(h) > 2n(h-2), n(h) > 4n(h-4), n(h) > 8n(n-6), … (by induction),

n(h) > 2in(h-2i)

Solving the base case we get: n(h) > 2 h/2-1

Taking logarithms: h < 2log n(h) +2

Thus the height of an AVL tree is O(log n)

3

4 n(1)

n(2)

© 2014 Goodrich, Tamassia, Goldwasser AVL Trees 4

InsertionInsertion is as in a binary search treeAlways done by expanding an external node.Example:

44

17 78

32 50 88

48 62

54w

b=x

a=y

c=z

44

17 78

32 50 88

48 62

before insertion

after insertion

© 2014 Goodrich, Tamassia, Goldwasser AVL Trees 5

Trinode RestructuringLet (a,b,c) be the inorder listing of x, y, z

Perform the rotations needed to make b the topmost node of the three

b=y

a=z

c=x

T0

T1

T2 T3

b=y

a=z c=x

T0 T1 T2 T3

c=y

b=x

a=z

T0

T1 T2

T3b=x

c=ya=z

T0 T1 T2 T3

Double rotation around c and a

Single rotation around b

© 2014 Goodrich, Tamassia, Goldwasser AVL Trees 6

Insertion Example, continued

88

44

17 78

32 50

48 62

2

5

1

1

3

4

2

1

54

1

T0T2

T3

x

y

z

2

3

4

5

6

7

1

88

44

17

7832 50

48

622

4

1

1

2 2

3

1

54

1

T0 T1

T2

T3

x

y z

unbalanced...

...balanced

1

2

3

4

5

6

7

T1

© 2014 Goodrich, Tamassia, Goldwasser AVL Trees 7

Restructuring (as Single Rotations)Single Rotations:

T0T1

T2

T3

c = xb = y

a = z

T0 T1 T2

T3

c = x

b = y

a = zsingle rotation

T3T2

T1

T0

a = x

b = y

c = z

T0T1T2

T3

a = xb = y

c = zsingle rotation

© 2014 Goodrich, Tamassia, Goldwasser AVL Trees 8

Restructuring (as Double Rotations)double rotations:

double rotationa = z

b = x

c = y

T0T2

T1

T3 T0

T2T3T1

a = zb = x

c = y

double rotationc = z

b = xa = y

T0T2

T1

T3 T0

T2T3 T1

c = zb = x

a = y

© 2014 Goodrich, Tamassia, Goldwasser AVL Trees 9

RemovalRemoval begins as in a binary search tree, which means the node removed will become an empty external node. Its parent, w, may cause an imbalance.Example:

44

17

7832 50

8848

62

54

44

17

7850

8848

62

54

before deletion of 32 after deletion

© 2014 Goodrich, Tamassia, Goldwasser AVL Trees 10

Rebalancing after a RemovalLet z be the first unbalanced node encountered while travelling up the tree from w. Also, let y be the child of z with the larger height, and let x be the child of y with the larger height

We perform a trinode restructuring to restore balance at z

As this restructuring may upset the balance of another node higher in the tree, we must continue checking for balance until the root of T is reached

44

17

7850

8848

62

54

w

c=x

b=y

a=z

44

17

78

50 88

48

62

54

© 2014 Goodrich, Tamassia, Goldwasser AVL Trees 11

AVL Tree PerformanceAVL tree storing n items

The data structure uses O(n) space

A single restructuring takes O(1) time

using a linked-structure binary tree

Searching takes O(log n) time

height of tree is O(log n), no restructures needed

Insertion takes O(log n) time

initial find is O(log n)

restructuring up the tree, maintaining heights is O(log n)

Removal takes O(log n) time

initial find is O(log n)

restructuring up the tree, maintaining heights is O(log n)

© 2014 Goodrich, Tamassia, Goldwasser

Java Implementation

AVL Trees 12

© 2014 Goodrich, Tamassia, Goldwasser

Java Implementation, 2

AVL Trees 13

© 2014 Goodrich, Tamassia, Goldwasser

Java Implementation, 3

AVL Trees 14