This document discusses balanced binary search trees (BSTs), specifically AVL trees. It begins by explaining that regular BSTs can have heights of O(N) in the worst case, making insertion and deletion operations slow. Balanced BSTs like AVL trees aim to keep the height O(log N) through rebalancing rotations after insertions or deletions. The document then covers AVL tree properties and balancing, the four types of rotations (LL, RR, LR, RL) used to rebalance after insertions, examples of constructing an AVL tree by insertion, and the different rotation types (R0, R1, R-1, L0, L1, L-1) used to rebalance

1. Data Structure and
Algorithm (CS-102)
Ashok K Turuk

2. Consider the insertion of following
element A, B, C, , ….,X, Y, Z into the
BST
A
B
C
X
Y
Z
O(N)

3. Consider the insertion of following
element Z, X, Y, , ….,C, B, A into the
BST
Z
X
Y
C
B
A
O(N)

4. 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.

5. 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

6. 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

7. AVL tree
TL 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

8. AVL Search Tree
C
G
D
A(0)
(0)
(-1)
(1)

9. 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

10. AVL Search Tree
(0)
(0)
(-1)
(1)
C
G
D
A
E

11. AVL Search Tree
(0)
(-1)
(-2)
(2)
C
G
D
A
E (0)

12. 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

13. 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 A
RR Rotation: Inserted node is in the right
sub-tree of right sub-tree of node A
LR Rotation: Inserted node is in the right
sub-tree of left sub-tree of node A
RL Rotation: Inserted node is in the left
sub-tree of right sub-tree of node A

14. LL Rotation
BL : Left Sub-tree of B
BR : Right Sub-tree of B
AR : Right Sub-tree of A
h : Height
A
B
c
AR
h
BL BR
(0)
(+1)
Insert X into BL
A
B
c
AR
h
BL BR
(+1)
(+2)
h+1
x
Unbalanced AVL
search tree after
insertion

15. LL Rotation
LL Rotation A
AR
c h
BR
(0)
B
BL
(0)
h+1
x
Balanced AVL
search tree after
rotation
A
B
c
AR
h
BL BR
(+1)
(+2)
h+1
x
Unbalanced AVL
search tree after
insertion

16. 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)

17. 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)

18. RR Rotation
A
B
c
AL
h
BL BR
(0)
(-1)
Insert X
into BR
h
A
B
c
BL BR
(-1)
(-2)
h+1
x
Unbalanced AVL
search tree after
insertion
AL

19. RR Rotation
RR Rotation B
A
c
BL BR
(0)
(0)
h+1
x
Balanced AVL
search tree after
Rotation
AL
A
B
c
BL BR
(-1)
(-2)
h+1
x
Unbalanced AVL
search tree after
insertion
AL
h

20. 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)

21. 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)

22. LR Rotation
Insert X
into CL
B
C
c
BL 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

23. LR Rotation
LR Rotation
B
C
c
BL CL CR
(-1)
(-1)
h
B
A
c
CL
CR
(-1)
(0)
x
Balanced AVL
search tree after
LR Rotation
BL
A
h
C(+2)
AR
(0)
AR
x

24. 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)

25. 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)

26. RL Rotation
Insert X
into CR
h
B
C
cAL
CL CR
(0)
(0)
h
A
(-1)
BR
h
B
C
cAL
CL CR
(-1)
(+1)
h
A
(-2)
x
Unbalanced AVL
search tree after
insertion
BR

27. RL Rotation
RL Rotation
h
B
C
cAL
CL CR
(-1)
(+1)
h
A
(-2)
BR
x
Balanced AVL search
tree after RL Rotation
x h
B
A
c
AL
CL
CR
(+1)
(0)
h
C
(0)
BR

28. 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)

29. 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

30. 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

31. Insert 64, 1
Insert 14
64
1
(+1)
(0)
14
64
1
(+2)
(-1)
(0)
64
(0)
14
1
(0)
(0)
LR

32. Insert 26, 13, 110,98
64
14
1
(0)
(0)
(0)
64
14
1
(-1)
(-1)
(-1)
98
110
26
13 (+1)
(0)
(0)
(0)

33. Insert 85
64
14
1
(-2)
(-1)
(-2)
98
110
26
13
(+2)
(0)
(+1)
(0)
(0)
85
64
14
1
(-1)
(-1)
(-1)
85
98
26
13
(0)
(0)
(0)
(0)
110
(0)
LL

34. Deletion in AVL search Tree
Deletion 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.

35. 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

36. 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.

37. R0 Rotation
A
B
c
AR
h
BL BR
(0)
(+1)
Delete node X
h
x
A
B
c
AR
h
BL BR
(0)
(+2)
Unbalanced AVL
search tree after
deletion of node
x
h -1

38. R0 Rotation
R0 Rotation
Balanced AVL
search tree after
rotation
A
B
c
AR
h
BL BR
(0)
(+2)
Unbalanced AVL
search tree after
deletion of x
A
AR
c h
BR
(+1)
B
BL
(-1)
BF(B) == 0, use
R0 rotation

39. 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

40. 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

41. R1 Rotation
A
B
c
AR
h -1
BL BR
(+1)
(+1)
Delete node X
h
x
h
A
B
c
AR
h
BL BR
(+1)
(+2)
Unbalanced AVL
search tree after
deletion of node
x
h -1
h -1

42. R1 Rotation
R1 Rotation
Balanced AVL
search tree after
rotation
A
B
c
AR
h -1
BL BR
(+1)
(+2)
A
AR
c h-1
BR
(0)
B
BL
(0)
BF(B) == 1, use
R1 rotation
h
h -1

43. 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)

44. 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)

45. R-1 Rotation
Delete X
B
C
c
BL 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

46. R-1 Rotation
R -1
B
C
c
BL CL CR
(0)
(-1)
h-1
A
h -1
(+2)
AR
C
B
A
c
CL
CR
(0)
(0)
Balanced AVL
search tree after
Rotation
BL
(0)
AR
h -1 h -1
BF(B) == -1,
use R-1 rotation

47. 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

48. 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

49. 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

50. L0 Rotation
A
B
c
AL
h
BL BR
(0)
(-1)
Delete X
h -1
B
A
c
BL
BR
(0)
(+1)
h
Balanced AVL
search tree after
deletion
AL
h-1
(-1)

51. 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)

52. 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-1
CR

53. 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-1
h-1

54. 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