Data structures trees and graphs

1. Data Structures
(AVL Tree)
Dr.N.S.Nithya
ASP/CSE

2. AVL Tree
 The AVL stands for Adelson-
Velskii and Landis, who are the
inventors of the AVL tree.AVL tree is a
self-balancing binary search tree in
which heights of the left sub trees and
right sub trees of any node differ by at
most one.

3. Balancing Factor
 Balance factor=height of left sub tree –
height of the right sub tree.
 The value of balance factor should always
be -1, 0 or +1.

4. Need for height balanced
trees
 If the binary search tree is either left skewed or
right skewed(not balanced) , the searching time
will be increased. Because searching time
depends upon the height of the tree. h=log(n)
where h is the height of the tree and n is the
number of nodes in the binary tree.It takes
log(n) for searching.

5. Balancing Factor Routine
int BF(node *T)
{
int lh,rh;
if(T==NULL)
return(0);
if(T->left==NULL)
lh=0;
else
lh=1+T->left->ht;
if(T->right==NULL)
rh=0;
else
rh=1+T->right->ht;
return(lh-rh);

6. Need for height balanced
trees
 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 Q(log N)
 Thus, our goal is to keep the height of a
binary search tree O(log N)
 Such trees are called balanced binary

7. How to balanace AVL tree (Rotation)
 An insertion or deletion may cause an
imbalance in an AVL tree.
 An AVL tree causes imbalance when
any of following condition occurs:
 i. An insertion into Right child’s right
subtree.
 ii. An insertion into Left child’s left subtree.
 iii. An insertion into Right child’s left
subtree.
 iv. An insertion into Left child’s right
subtree.

8. How to balanace AVL tree (Rotation)
 Whenever the tree becomes imbalanced due to
any operation we use rotation operations to
make the tree balanced.
 Rotation is the process of moving nodes
either to left or to right to make the tree
balanced.

9. Single Left Rotation (LL Rotation)
 If a tree becomes unbalanced, when a node is
inserted into the right subtree of the right
subtree, then we perform a single left rotation .
 If BF(node) = +2 and BF(node -> left-child) = +1, perform LL
rotation.

10. Single Right Rotation (RR Rotation)
 AVL tree may become unbalanced, if a node
is inserted in the left subtree of the left
subtree. The tree then needs a right rotation.
 If BF(node) = -2 and BF(node -> right-child) = 1, perform RR
rotation.


11. node * rotateright(node *x)
{
node *y;
y=x->left;
x->left=y->right;
y->right=x;
x->ht=height(x);
y->ht=height(y);
return(y);
}
node * rotateleft(node *x)
{
node *y;
y=x->right;
x->right=y->left;
y->left=x;
x->ht=height(x);
y->ht=height(y);
return(y);
}

12. Left Right Rotation (LR Rotation)
(Double Rotation)
 AVL tree may become unbalanced, if a node is inserte
the left subtree of the right subtree .The tree needs
Rotation is a sequence of single left rotation followed
single right rotation.
 If BF(node) = -2 and BF(node -> right-child) = +1, perform RL rotation.

13. Right Left Rotation (RL Rotation)
(Double Rotation)
 AVL tree may become unbalanced, if a node is
inserted in the right subtree of the left subtree
.The tree needs The RL Rotation is sequence
of single right rotation followed by single left
rotation.
 If BF(node) = +2 and BF(node -> left-child) = -1, perform LR
rotation.

14. node * RR(node *T) //insert right subtree of right
{
T=rotateleft(T);
return(T);
}
node * LL(node *T) //insert left subtree of left
{
T=rotateright(T);
return(T);
}
node * LR(node *T) // insert left subtree of right
{
T->left=rotateleft(T->left);
T=rotateright(T);
return(T);
}
node * RL(node *T) //insert right subtree of left
{
T->right=rotateright(T->right);
T=rotateleft(T);
return(T);
}

15. Left Rotation
Right Rotation

16. Left-Right Rotation

17. Right-Left Rotation

18. Operations on an AVL Tree
 The following operations are performed
on AVL tree...
 Search
 Insertion
 Deletion
 search operation is similar to Binary
search tree.

19. Insertion Operation in AVL Tree
 Step 1 - Insert the new element into the
tree using Binary Search Tree insertion
logic.
 Step 2 - After insertion, check
the Balance Factor of every node.
 Step 3 - If the Balance Factor of every
node is 0 or 1 or -1 then go for next
operation.
 Step 4 - If the Balance Factor of any
node is other than 0 or 1 or -1 then that
tree is said to be imbalanced. In this
case, perform suitable Rotation to make
it balanced and go for next operation.

20. Rotating conditions
 If BF(node) = +2 and BF(node -> left-child) =
+1, perform LL rotation.
 If BF(node) = -2 and BF(node -> right-child) =
1, perform RR rotation.
 If BF(node) = -2 and BF(node -> right-child) =
+1, perform RL rotation.
 If BF(node) = +2 and BF(node -> left-child) =
1, perform LR rotation.

21. An Extended Example
Insert 3,2,1,4,5,6,7, 16,15,14
3
Fig 1
3
2
Fig 2
3
2
1
Fig 3
2
1
3
Fig 4
2
1
3
4
Fig 5
2
1
3
4
5
Fig 6
Single rotation
Single rotation
Insert 3 Insert 2
Insert 1
Insert 4
Insert 5
0
1
0
1
0
2 0
0 0
-
1
-1
0
0
-
2
-
2
-
1 0
0

22. 2
1
4
5
3
Fig 7 6
2
1
4
5
3
Fig 8
4
2
5
6
1 3
Fig 9
4
2
5
6
1 3
7
Fig 10
4
2
6
7
1 3
5
Fig 11
Single rotation
Single rotation
Insert 6
Insert 7
-1
0
0
0
0
0
0
0
-1
-
1
-
2
0
-
1
0
0
0
0
-
1
0
0
0
-
2
-
1
0
0
1
0
0
0
0
0
0

23. 4
2
6
7
1 3
5 16
Fig 12
4
2
6
7
1 3
5 16
15
Fig 13
4
2
6
15
1 3 5
16
7
Fig 14
Double rotation
Insert 16
Insert 15
-1
-
1
-1
0
0
0
0
0
-
2
-
1
-2
-
1
0
1
0
0
0
-
1
0
0
0
-
1
0
0
0
0

24. 5
4
2 7
15
1 3 6
16
14
Fig 16
4
2
6
15
1 3 5
16
7
14
Fig 15
Double rotation
Insert 14
-
2
-2
1
0
0
-1
0
0
0
0
-
1
0
0
0
0
1
0
0 0
0
1

25. Operations on an AVL Tree
 The following operations are performed
on AVL tree...
 Search
 Insertion
 Deletion
 search operation is similar to Binary
search tree.

26. node * insert(node *T,int x)
{
if(T==NULL)
{
T=(node*)malloc(sizeof(node));
T->data=x;
T->left=NULL;
T->right=NULL;
}
else
if(x > T->data) // insert in right subtree
{
T->right=insert(T->right,x);
if(BF(T)==-2)
if(x>T->right->data)
T=RR(T);
else
T=RL(T);
}

27. else
if(x<T->data)
{
T->left=insert(T->left,x);
if(BF(T)==2)
if(x < T->left->data)
T=LL(T);
else
T=LR(T);
}
T->ht=height(T);
return(T);
}

28. Deletion in AVL Trees
 Step 1: Find the element in the tree.
 Step 2: Delete the node, as per the
BST Deletion.
 Step 3: Two cases are possible:-
 Case 1: Deleting from the right subtree.
 Case 2: Deleting from left subtree.

29. Case 1: Deleting from the right
subtree.
 1A. If BF(node) = +2 and BF(node ->
left-child) = +1, perform LL rotation.
 1B. If BF(node) = +2 and BF(node ->
left-child) = -1, perform LR rotation.
 1C. If BF(node) = +2 and BF(node ->
left-child) = 0, perform LL rotation.

30. Case 2: Deleting from left subtree.
 2A. If BF(node) = -2 and BF(node ->
right-child) = -1, perform RR rotation.
 2B. If BF(node) = -2 and BF(node ->
right-child) = +1, perform RL rotation.
 2C. If BF(node) = -2 and BF(node ->
right-child) = 0, perform RR rotation.

31. AVL Tree Example:
• Now remove 53(leaf Node)
14
17
7
4
53
11
12
8 13

32. AVL Tree Example:
• Now remove 53, unbalanced
14
17
7
4
11
12
8 13

33. AVL Tree Example:
• Balanced! Remove 11(node with two
children)
14
17
7
4
11
12
8
13

34. AVL Tree Example:
• Remove 11, replace it with the largest in its left
branch tree balanced
14
17
7
4
8
12
13

35. AVL Tree Example:
• Remove 8(place with inorder predessor)replace
with largest element in the left subtree, unbalanced
14
17
4
7
12
13

36. AVL Tree Example:
• Remove 8, unbalanced
14
17
4
7
12
13

37. AVL Tree Example:
• Balanced!!
14
17
4
7
12
13

38. node * Delete(node *T,int x)
{
node *p;
if(T==NULL)
{
return NULL;
}
else
if(x > T->data) // delete in right
subtree
{
T->right=Delete(T->right,x);
if(BF(T)==2)
if(BF(T->left)>=0)
T=LL(T);
else

39. if(x<T->data)
{
T->left=Delete(T->left,x);
if(BF(T)==-2) //Rebalance
during windup
if(BF(T->right)<=0)
T=RR(T);
else
T=RL(T);
}

40. else
{
//data to be deleted is found
if(T->right!=NULL)
{ //delete its inorder succesor
p=p->right;
while(p->left!= NULL)
p=p->left;
T->data=p->data;
T->right=Delete(T->right,p->data);
if(BF(T)==2)//Rebalance during windup
if(BF(T->left)>=0)
T=LL(T);
else
T=LR(T);
}
else
return(T->left);
}
T->ht=height(T);
return(T);
}

41. Height calculation
int height(node *T)
{
int lh,rh;
if(T==NULL)
return(0);
if(T->left==NULL)
lh=0;
else
lh=1+T->left->ht;
if(T->right==NULL)
rh=0;
else
rh=1+T->right->ht;
if(lh>rh)
return(lh);
return(rh);

42. Applications of AVL Tree
 AVL trees are used for frequent insertion.
 It is used in Memory management
subsystem of linux kernel to search
memory regions of processes during
preemption.
 Sort the following database:
 Dictioonary
 Google search engine
 some sites to take data faster
 sites which have large amount of data
example is Facebook.

43. In Class Exercises
 Build an AVL tree with the following values:
15, 20, 24, 10, 13, 7, 30, 36, 25

44. 15
15, 20, 24, 10, 13, 7, 30, 36, 25
20
24
15
20
24
10
13
15
20
24
13
10
13
20
24
15
10

45. 13
20
24
15
10
15, 20, 24, 10, 13, 7, 30, 36, 25
7
13
20
24
15
10
7
30
36
13
20
30
15
10
7
36
24

46. 13
20
30
15
10
7
36
24
15, 20, 24, 10, 13, 7, 30, 36, 25
25
13
20
30
15
10
7
36
24
25
13
24
36
20
10
7
25
30
15

47. Remove 24 and 20 from the AVL tree.
13
24
36
20
10
7
25
30
15
13
20
36
15
10
7
25
30
13
15
36
10
7
25
30
13
30
36
10
7
25
15