Data Structure and Algorithm (CS-102)

Ashok K Turuk

Consider the insertion of following element A, B, C, , ….,X, Y, Z into the BST

AB

C

X

Y

Z

O(N)

Consider the insertion of following element Z, X, Y, , ….,C, B, A into the BST

Z

X

Y

C

B

A

O(N)

Balanced binary tree

• The disadvantage of a binary search tree is that its height can be as large as N-1

• This means that the time needed to perform insertion and deletion and many other operations can be O(N) in the worst case

• We want a tree with small height• A binary tree with N node has height at least (log

N) • Thus, our goal is to keep the height of a binary

search tree O(log N)• Such trees are called balanced binary search

trees. Examples are AVL tree, red-black tree.

AVL tree

Height of a node• The height of a leaf is 1. The height

of a null pointer is zero.• The height of an internal node is the

maximum height of its children plus 1

AVL tree

• An AVL tree is a binary search tree in which– for every node in the tree, the height of

the left and right subtrees differ by at most 1.

– An empty binary tree is an AVL tree

AVL treeTL left subtree of T h(TL ) Height of the subtree TL

TR Right subtree of T h(TR ) Height of the subtree TR

T is an AVL tree iff TL and TR are AVL tree and |h(TL ) - h(TR ) | <= 1

h(TL ) - h(TR ) is known as balancing factor (BF) and for an AVL tree the BF of a node can be either 0 , 1, or -1

AVL Search Tree

C

G

D

A(0)

(0)

(-1)

(1)

Insertion in AVL search Tree

Insertion into an AVL search tree may affect the BF of a node, resulting the BST unbalanced.

A technique called Rotation is used to restore he balance of the search tree

AVL Search Tree

(0)

(0)

(-1)

(1)

C

G

D

A

E

AVL Search Tree

(0)

(-1)

(-2)

(2)

C

G

D

A

E (0)

Rotation

To perform rotation – Identify a specific node A whose BF(A) is neither 0, 1, or -1 and which is the nearest ancestor to the inserted node on the path from the inserted node to the root

Rotation Rebalancing rotation are classified as LL,

LR, RR and RL LL Rotation: Inserted node is in the left

sub-tree of left sub-tree of node ARR Rotation: Inserted node is in the

right sub-tree of right sub-tree of node ALR Rotation: Inserted node is in the right

sub-tree of left sub-tree of node ARL Rotation: Inserted node is in the left

sub-tree of right sub-tree of node A

LL Rotation

BL : Left Sub-tree of BBR : Right Sub-tree of BAR : Right Sub-tree of Ah : Height

A

B

c

AR

hBL BR

(0)

(+1)

Insert X into BL

A

B

c

AR

hBL BR

(+1)

(+2)

h+1 x

Unbalanced AVL search tree after insertion

LL Rotation

LL Rotation

A

ARc h

BR

(0)B

BL

(0)

h+1 x

Balanced AVL search tree after rotation

A

B

c

AR

hBL BR

(+1)

(+2)

h+1 x

Unbalanced AVL search tree after insertion

LL Rotation Example

Insert 36(0)

(+1)

96

85 110

64 90 (0)

(0)

(0)

(+1)

(+2)

96

85 110

64 90 (0)

(0)

(+1)

36

Unbalanced AVL search tree

(0)

LL Rotation Example

LL Rotation

(+1)

(+2)

96

85 110

64 90 (0)

(0)

(+1)

36

Unbalanced VAL search tree

(0)

85

110

64

(0)

(+1) 96

(0)

(0)

Balanced AVL search tree after LL rotation

(0)

9036(0)

RR Rotation

A

B

c

AL

h

BL BR

(0)

(-1)

Insert X into BR

h

A

B

c

BL BR

(-1)

(-2)

h+1x

Unbalanced AVL search tree after insertion

AL

RR Rotation

RR Rotation

B

Ac

BL BR

(0)

(0)

h+1x

Balanced AVL search tree after Rotation

AL

A

B

c

BL BR

(-1)

(-2)

h+1x

Unbalanced AVL search tree after insertion

AL

h

RR Rotation Example

Insert 65(0)

(-1)

34

26 44

40 56(0)

(0)

(0)

(0)

(-2)

34

26 44

40 56(-1)

(-1)

(0)

65

Unbalanced AVL search tree

(0)

RR Rotation Example

RR Rotation (0)

(-2)

44

34 56

40 65(0)

(-1)

(0)26

Balanced AVL search tree after RR rotation

(0)

(0)

(-2)

34

26 44

40 56(-1)

(-1)

(0)

65(0)

LR Rotation

Insert X into CL

B

C

c

BL

h-1

CL CR

(0)

(-1)

h

B

C

c

CL CR

(-1)

(-1)

x

Unbalanced AVL search tree after insertion

BL

A

h

A(+1)

AR

(+2)

AR

LR Rotation

LR Rotation

B

C

c

BL

h

CL CR

(-1)

(-1)

h

BA

cCL

CR

(-1)(0)

x

Balanced AVL search tree after LR Rotation

BL

A

h

C(+2)

AR

(0)

AR

x

LR Rotation Example

Insert 37(0)

(+1)

44

30 76

16 39 (0)

(0)

(0)

(-1)

(+2)

44

30 76

16 39 (+1)

(0)

(+1)

37

Unbalanced AVL search tree

(0)

LR Rotation Example

LR Rotation

(-1)

(+2)

44

30 76

16 39 (+1)

(0)

(0)

(0)

(0)

39

30 44

16 37 (0)

(-1)

(0)

76

Balanced AVL search tree

(0)37

(0)

RL Rotation

Insert X into CR

h

B

C

cAL h-1

CL CR

(0)

(0)

h

A(-1)

BR

h

B

C

cAL h-1

CL CR

(-1)

(+1)

h

A(-2)

x

Unbalanced AVL search tree after insertion

BR

RL Rotation

RL Rotation

h

B

C

cAL h-1

CL CR

(-1)

(+1)

h

A(-2)

BR

x

Balanced AVL search tree after RL Rotation

x h

BA

c

AL

h-1

CLCR

(+1)

(0)

h

C(0)

BR

RL Rotation Example

Insert 41(0)

(-1)

34

26 44

40 56(0)

(0)

(0)

(0)

(-2)

34

26 44

40 56(0)

(+1)

(-1)

41

Unbalanced AVL search tree

(0)

RL Rotation Example

RL Rotation

(0)

(-2)

34

26 44

40 56(0)

(+1)

(-1)

(0)

(0)

(+1)

(0)

40

34 44

41 56(0)

(0)

(0)

Balanced AVL search tree

26

41

AVL Tree

Construct an AVL search tree by inserting the following elements in the order of their occurrence

64, 1, 14, 26, 13, 110, 98, 85

Insert 64, 1

Insert 14

64

1

(+1)

(0)

14

64

1

(+2)

(-1)

(0)

64

(0)14

1

(0)

(0)LR

Insert 26, 13, 110,98

64

14

1

(0)

(0) (0)

64

14

1

(-1)

(-1) (-1)

98

11026

13 (+1)(0)

(0)

(0)

Insert 85

64

14

1

(-2)

(-1) (-2)

98

11026

13(+2)

(0)

(+1)

(0)

(0)85

64

14

1

(-1)

(-1) (-1)

85

9826

13(0)

(0)

(0)

(0)

110

(0)

LL

Deletion in AVL search TreeDeletion in AVL search tree proceed

the same way as the deletion in binary search tree

However, in the event of imbalance due to deletion, one or more rotation need to be applied to balance the AVL tree.

AVL deletion

Let A be the closest ancestor node on the path from X (deleted node) to the root with a balancing factor +2 or -2

Classify the rotation as L or R depending on whether the deletion occurred on the left or right subtree of A

AVL Deletion

Depending on the value of BF(B) where B is the root of the left or right subtree of A, the R or L imbalance is further classified as R0, R1 and R -1 or L0, L1 and L-1.

R0 Rotation

A

B

c

AR

hBL BR

(0)

(+1)

Delete node X

hx

A

B

c

AR

hBL BR

(0)

(+2)

Unbalanced AVL search tree after deletion of node x

h -1

R0 Rotation

R0 Rotation

Balanced AVL search tree after rotation

A

B

c

AR

hBL BR

(0)

(+2)

Unbalanced AVL search tree after deletion of x

A

ARc h

BR

(+1)B

BL

(-1)

BF(B) == 0, use R0 rotation

R0 Rotation Example

Delete 60

Unbalanced AVL search tree after deletion

(0)

(0)

(+1)

46

20 54

18 23 (-1)

(-1)

(+1)

7(0) (0)

60

24

(0)

(+2)

46

20 54

18 23 (-1)

(0)

(+1)

7(0) (0)

24

R0 Rotation Example

R0(0)

(+2)

46

20 54

18 23 (-1)

(0)

(+1)

7(0) (0)

24

Balanced AVL search tree after deletion

(+1)

(-1)

20

18 46

7 23(-1)

(+1)

(0)

54(0)

(0)

24

R1 Rotation

A

B

c

AR

h -1BL BR

(+1)

(+1)

Delete node X

hxh

A

B

c

AR

h

BL BR

(+1)

(+2)

Unbalanced AVL search tree after deletion of node x

h -1

h -1

R1 Rotation

R1 Rotation

Balanced AVL search tree after rotation

A

B

c

AR

h -1BL BR

(+1)

(+2)

A

ARc h-1

BR

(0)B

BL

(0)

BF(B) == 1, use R1 rotation

h

h -1

R1 Rotation Example

Delete 39

Unbalanced AVL search tree after deletion

(0)

(+1)

(+1)

37

26 41

18 28(0)

(+1)

(+1)

16(0)

39

(+1)

(+2)

37

26 41

18 28 (0)

(0)

(+1)

16(0)

R1 Rotation Example

R1 Rotation

Balanced AVL search tree after deletion

(+1)

(+2)

37

26 41

18 28 (0)

(0)

(+1)

16(0)

(+1)

(0)

26

18 37

16 28 (0)

(0)

(0)41

(0)

R-1 Rotation

Delete X

B

C

c

BL

h-1

CL CR

(0)

(-1)

h-1

A

h

(+1)

AR

x

A

B

C

c

CL CR

(0)

(-1)

Unbalanced AVL search tree after deletion

BL

(+2)

AR

h-1

R-1 Rotation

R -1

B

C

c

BL

h-1

CL CR

(0)

(-1)

h-1

A

h -1

(+2)

AR

C

BA

c

CLCR

(0)

(0)

Balanced AVL search tree after Rotation

BL

(0)

AR

h -1 h -1

BF(B) == -1, use R-1 rotation

R-1 Rotation Example

Delete 52(-1)

(+1)

44

22 48

18 28 (0)

(-1)

(0)52

2923

(-1)

(+2)

44

22 48

18 28 (0)

(0)

(0)

23

Unbalanced AVL search tree after deletion

(0) 29

R-1 Rotation Example

R-1(-1)

(+1)

44

22 48

18 28 (0)

(-1)

(0)52

2923

(0)

(0)

28

22

4818 23(0)

(0)

(0)

44

Balanced AVL search tree after rotation

(0)

29

L0 Rotation

A

B

c

AL

h

BL BR

(0)

(-1)

Delete X

h

x

A

B

c

BL BR

(0)

(-2)

h

Unbalanced AVL search tree after deletion

AL

h-1

L0 Rotation

A

B

c

AL

h

BL BR

(0)

(-1)

Delete X

h -1

B

Ac

BL

BR

(0)

(+1)

h

Balanced AVL search tree after deletion

AL

h-1

(-1)

L1 Rotation

A

B

c

AL

h-1

CL

BR

(+1)

(-1)

Delete X

h

x C

CR

h-1

(0)

A

B

c

CL

BR

(+1)

(-2)

h-1

Unbalanced AVL search tree after deletion

AL

h-1

C

CR

(0)

L1 Rotation

A

B

c

AL

h-1

CL

BR

(+1)

(-2)

L1

h-1

C

CR

h-1

(0)

(0)

A

C

B

c

CL

BR

(0)

(0)

h-1

Unbalanced AVL search tree after deletion

AL

h-1CR

L-1 Rotation

A

B

c

AL

h

BL BR

(-1)

(-1)

Delete X

h

x

h-1

A

B

c

BL BR

(-1)

(-2)

h

Unbalanced AVL search tree after deletion

AL

h-1h-1

L-1 Rotation

A

B

c

AL

h

BL BR

(-1)

(-2)

Delete X

h-1

B

A

c

BL BR

(-1)

(0)

h

Balanced AVL search tree after deletion

AL

h-1

h-1