The document is a lecture on AVL trees, explaining their structure and balance compared to binary search trees. It details the concept of balance factors, rotations necessary for maintaining tree balance during insertions or deletions, and provides examples of each case for balancing. The lecture is part of a series from the International Islamic University in Islamabad.

1. International Islamic University H-10, Islamabad, Pakistan
Data Structure
Lecture No. 17
AVL Tree
Engr. Rashid Farid Chishti
http://youtube.com/rfchishti
http://sites.google.com/site/chishti
Video Lecture

2.  BST for 3 4 5 7 9 14 15 16 17 20
Binary Search Tree
3
20
17
16
15
14
9
7
5
4
Linked List!

3.  AVL (Adelson-Velskii and Landis) tree.
 An AVL tree is identical to a Binary Search Tree except
 height of the left and right subtrees can differ by 0, -1, or 1.
 height of an empty tree is defined to be (0).
AVL Tree
An AVL Tree Not an AVL tree
Level 0
Level 2
Level 1
Level 3
14
15
4
9 18
16 20
14
15
4
18
16 20

4.  The height of a binary tree is the maximum level of its leaves.
 Balance Factor = height of its left subtree - height of its right subtree.
 A balance factor is stored at every node.
 Here, for example, in a balanced tree. Each node has an indicated balance
factor of 1, 0, or –1.
AVL Tree: Balance Factor
Balanced
Trees
3
b = 0
h = 1
9
b = 0
h = 1
b = 1 – 1 = 0
h = 2
4
b = 0 – 2 = – 2
h = 3
15
14
b = 2 – 3 = -1
h = 4
b = 0
h = 2
18
14
b = 0
h = 1
15
14
4
b = 0
h = 2
h = 1
b = 0
h = 1
b = 0
14
4
b = 1
h = 2
15
14
b = -1
h = 2
16
b = 0
h = 1
20
b = 0
h = 1

5.  Insertion and removal may change the balance condition of a tree
 A rotation is required to restore the balance condition
 There are two types of rotations
 Left Rotation
 Right Rotation
AVL Tree: Rotations

6.  Let U be the nearest imbalanced node to the inserted one
 Case 1 (L-L): Rotate Right Node U
 Case 2 (R-R): Rotate Left Node U
 Case 3 (R-L): Rotate Right then Rotate Left
 Case 4 (L-R): Rotate Left then Rotate Right
 In cases 1 and 2, use single rotation to restore the tree
 In cases 2 and 3, use double rotation to restore the tree
AVL Tree: Imbalance

7.  Case 1 (L-L):
AVL Tree: Case 1 ( Left Left )
Rotate as
Right Child
b = 2-0 = 2
h = 3
h = 1
h = 2
left
left
2
3
Inset(3) Inset(2) Inset(1)
Rotate Right Node No. 3
1 2.5
2.5
h = 1
b = 1 – 1 = 0
h = 2
3
1
h = 1
2
Balance Factor = height of its left subtree - height of its right subtree.
h = 2
h = 1
b = 1-2 = -1
h = 3
The height of a binary tree is the maximum level of its leaves.

8.  Case 2 (R-R):
AVL Tree: Case 2 ( Right Right )
b = -2
h = 3
3
h = 2
right
right
Inset(1) Inset(2) Inset(3)
Rotate as
Left Child
Rotate Left Node No. 1
1.5
1
2
h = 1
Balance Factor = height of its left subtree - height of its right subtree.
The height of a binary tree is the maximum level of its leaves.
1.5
h = 1
b = 1 – 1 = 0
h = 2
3
1
h = 1
2
h = 2
h = 1
b = 2-1 = 1
h = 3

9.  Case 3 (R-L):
AVL Tree: Case 3 ( Right Left ) Step 1
h = 2
h = 1
b = 0 – 2 = -2
h = 3
left
right
1
Inset(1) Inset(3) Inset(2)
Step 1: Rotate Right Node No. 3
Rotate as
Right Child
3
2.5
2
2.5
h = 1
1
b = 0 – 2 = -2
h = 3
h = 2
2
right
right
3
Balance Factor = height of its left subtree - height of its right subtree.
The height of a binary tree is the maximum level of its leaves.

10.  Case 3 (R-L):
AVL Tree: Case 3 ( Right Left ) Step 2
b = 0 – 2 = - 2
h = 3
3
h = 1
h = 2
right
right
1
Rotate as
Left Child
Step 2: Rotate Left Node No. 1
3
h = 1
h = 2
1
b = 1-1 = 0
2
1.5
1.5
2
Balance Factor = height of its left subtree - height of its right subtree.
The height of a binary tree is the maximum level of its leaves.

11.  Case 4 (L-R):
AVL Tree: Case 4 ( Left Right ) Step 1
b = 2 – 0 = 2
h = 3
h = 1
h = 2
right
left
3
1
Inset(3) Inset(1) Inset(2)
Rotate Left Node No. 1
Rotate as
Left Child
b = 2 - 0 = 2
h = 3
1
h = 1
h = 2
left
left
2
3
1.5
2
1.5
Balance Factor = height of its left subtree - height of its right subtree.
The height of a binary tree is the maximum level of its leaves.

12.  Case 4 (L-R):
AVL Tree: Case 4 ( Left Right ) Step 2
2
3
h = 1
1
h = 1
b = 1 - 1 = 0
h = 2
Rotate Right Node No. 3
b = 2 - 0 = 2
h = 3
1
h = 1
h = 2
left
left
2
Rotate as
Right Child
3
Balance Factor = height of its left subtree - height of its right subtree.
The height of a binary tree is the maximum level of its leaves.
2.5
2.5

13.  Insert(3)
AVL Tree Building Example
3
h = 1, b = 0

14.  Insert(2)
AVL Tree Building Example
2
3
h = 2, b = 1
h = 1

15.  Insert(1)
AVL Tree Building Example
2
1
Case 1: L-L
3
h = 2
h = 1
h = 3, b = 2 – 0 = 2
 Case 1: L-L
2.5
3
1
2
Rotate as
Right Child
left
left
2
3
1 2.5

16. AVL Tree Building Example
1
2
3
h = 1 h = 1
h = 2
 Case 1: L-L
2.5
3
1
2
Rotate as
Right Child
left
left
2
3
1 2.5

17.  Insert(4)
AVL Tree Building Example
1
2
3
4
h = 1
h = 1
h = 2
h = 1
h = 2
h = 3 , b = 1 – 2 = -1

18.  Insert(5)
AVL Tree Building Example
1
Case 2: R-R
3
2
5
h = 1
h = 2
h = 1
h = 1
h = 2
h = 3
h = 3 , b = 0 – 2 = -2
4
 Case 2: R-R
3
Rotate
as Left
Child
1.5
1
2
1.5
3
1
2
right
right

19. AVL Tree Building Example
1
2
5
3
4
 Case 2: R-R
3
Rotate
as Left
Child
1.5
1
2
2.5
3
1
2
right
right
h = 1
h = 1
h = 1
h = 2
h = 3

20.  Insert(6)
AVL Tree Building Example
1 4
6
Case 2: R-R
5
3
h = 1
h = 3
2
h = 1
h = 1 h = 2
h = 1
h = 2
h = 3
h = 4, b = 1 – 3 = -2
 Case 2: R-R
3
Rotate
as Left
Child
1.5
1
2
2.5
3
1
2
right
right
3

21. AVL Tree Building Example
5
2
6
1 3
4
h = 2
h = 1 h = 1
 Case 2: R-R
3
Rotate
as Left
Child
1.5
1
2
2.5
3
1
2
right
right
h = 1
h = 2
h = 3

22.  Insert(7)
AVL Tree Building Example
5
2
6
4
7
3
1
h = 1
h = 2
h = 1 h = 1 h = 1
h = 2
h = 3
h = 2
h = 3, b = -2
Case 2: R-R
 Case 2: R-R
3
Rotate
as Left
Child
1.5
1
2
2.5
3
1
2
right
right

23. AVL Tree Building Example
2
7
6
3
1
4
5
h = 2
h = 1 h = 1 h = 1
 Case 2: R-R
3
Rotate
as Left
Child
1.5
1
2
1.5
3
1
2
right
right
h = 1
h = 2
h = 3

24.  Insert(16)
AVL Tree Building Example
2
7
6
3
1
4
5
16
h = 1
h = 2
h = 1 h = 1 h = 1 h = 1
h = 2
h = 3
h = 2
h = 3, b = 1 - 2 = 1
h = 4, b = 2 – 3 = – 1

25.  Insert(15)
AVL Tree Building Example
2
7
6
3
1
4
5
16
Case 3: R-L
15
h = 1
h = 1
h = 2
h = 1 h = 1 h = 1
h = 4
h = 3
h = 2
h = 2
h = 3, b = -2
 Case 3: R-L
3
1
2
left
right
1
3
2

26. AVL Tree Building Example
2 6
3
1
4
5
16
7
15
h = 2
h = 1 h = 1 h = 1
 Case 3: R-L
3
1
2
left
right
1
3
2
h = 4
h = 1
h = 1
h = 2
h = 3

27.  Insert(14)
AVL Tree Building Example
2 6
3
1
4
16
15
5
Case 3: R-L
14
h = 2
h = 1 h = 1 h = 1
h = 4
h = 3
7
h = 1
h = 2
h = 1
h = 1
h = 2
h = 3
h = 4, b = 1 – 3 = – 2
Case 3: R-L
1
2
left
right
1
3
2.5
2.5
3
1.5
2
1.5

28. AVL Tree Building Example
2 7
3
1
4
16
14
6
5
15
h = 2
h = 1 h = 1 h = 2
h = 4
h = 3
h = 1
h = 2
Case 3: R-L
1
2
left
right
1
3
2.5
2.5
3
1.5
2
1.5
h = 1
h = 1

29.  Insert(13)
AVL Tree Building Example
2 7
3
4
16
15
6
5 14
13
Case 2: R-R
1
h = 1
h = 1
h = 1
h = 4, b = 2 – 3 = -1
h = 2
h = 1 h = 1 h = 2
h = 4
h = 3
h = 1
h = 2
h = 2
h = 3
h = 5, b = 2 – 4 = -2
Case 2: R-R
1
2
3
1.5
right
right
1
1.5 3
2

30. AVL Tree Building Example
4 15
7
16
14
13
2
1 3
h = 3
h = 1 h = 1
h = 2
6
5
h = 1
h = 2 h = 2
h = 4
h = 3
h = 1
Case 2: R-R
1
2
3
1.5
right
right
1
1.5 3
2
h = 1

31.  Insert(12)
AVL Tree Building Example
4 15
7
16
14
2
1 3
6
5
12
13
Case 1: L-L
h = 3
h = 1 h = 1 h = 1
h = 2 h = 2
h = 1
h = 2
h = 4
h = 3
h = 1
h = 1
h = 2
h = 3, b = 2

32. AVL Tree Building Example
4 15
7
16
13
2
1 3
6
5 12 14
h = 3
h = 1 h = 1 h = 1
h = 2 h = 2 h = 2
h = 4
h = 3
h = 1
h = 1 h = 1

33.  Insert(11)
AVL Tree Building Example
4 15
7
16
13
2
1 3
6
5 14
12
11
h = 1
Case 1: L-L
h = 3
h = 1 h = 1 h = 1
h = 2 h = 2
h = 1
h = 2
h = 4
h = 3
h = 1
h = 1
h = 2
h = 3
h = 4, b = 2

34. AVL Tree Building Example
4 13
7
12
2
1 3
6
5 11 16
15
14
h = 3
h = 1 h = 1 h = 1
h = 2 h = 2 h = 2
h = 4
h = 3
h = 1
h = 2
h = 1
h = 1

35.  Insert(10)
AVL Tree Building Example
4 13
7
15
12
2
1 3
6
5 14 16
10
11
Case 1: L-L
h = 3
h = 1 h = 1 h = 1
h = 2 h = 2
h = 1
h = 2
h = 4
h = 3
h = 1
h = 2
h = 1 h = 1
h = 2
h = 3, b = -2

36. AVL Tree Building Example
4 13
7
15
11
2
1 3
6
5 14 16
10 12
h = 3
h = 1 h = 1 h = 1
h = 2 h = 2 h = 2
h = 4
h = 3
h = 1
h = 2
h = 1
h = 1 h = 1

37.  Insert(8)
AVL Tree Building Example
4 13
7
15
11
2
1 3
6
5 14 16
12
10
8
h = 3
h = 1 h = 1 h = 1
h = 2 h = 2
h = 1
h = 2
h = 4
h = 3
h = 1
h = 2
h = 2
h = 1 h = 1 h = 1
h = 3
h = 4
h = 5

38.  Insert(9)
AVL Tree Building Example
4 13
7
15
11
2
1 3
6
5 14 16
12
10
9
8
Case 4: L-R
h = 3
h = 1 h = 1 h = 1
h = 2 h = 2
h = 1
h = 3
h = 5
h = 4
h = 1
h = 2
h = 1
h = 1
h = 1
h = 2
h = 2
h = 3, b = 2

39.  Insert(9)
AVL Tree Building Example
4 13
7
15
11
2
1 3
6
5 14 16
12
9
8 10
h = 3
h = 1 h = 1 h = 1
h = 2 h = 2
h = 1
h = 3
h = 4
h = 3
h = 1
h = 2
h = 1
h = 1
h = 2
h = 1

40. #include<iostream>
#include<stdlib.h>
using namespace std;
struct Node {
int data;
Node *left;
Node *right;
int h;
};
// An AVL tree node
class AVL_Tree {
Node *root;
int Max (int a, int b);
int Height (Node* N);
Node* New_Node (int key);
Node* Right_Rotate(Node* y);
Node* Left_Rotate (Node* x);
int Get_Balance (Node* N);
40
Example 1: Implementation of AVL Tree
1

41. void Pre_Order (Node* node);
Node* Insert (Node* node, int key);
public:
AVL_Tree ( ) {root = NULL;}
void Pre_Order ( );
void Insert (int key);
};
// A utility function to get maximum of two integers
int AVL_Tree :: Max(int a, int b) {
return (a > b)? a : b;
}
int AVL_Tree :: Height(Node *N) {
if (N == NULL)
return 0;
return N->h;
}
41
Example 1: Implementation of AVL Tree
2

42. Node* AVL_Tree :: New_Node(int key) {
Node* node = new Node();
node->data = key;
node->left = NULL;
node->right = NULL;
node->h = 1; // new node is initially added at leaf
return(node);
}
// A utility function to right rotate subtree rooted with y
Node* AVL_Tree :: Right_Rotate(Node *y) {
Node *x = y->left; Node *T2 = x->right;
x->right = y; y->left = T2; // Perform rotation
y->h = Max(Height(y->left), Height(y->right)) + 1; // Update heights
x->h = Max(Height(x->left), Height(x->right)) + 1;
return x; // Return new root
}
42
Example 1: Implementation of AVL Tree
3

43. // A utility function to left rotate subtree rooted with x
Node* AVL_Tree :: Left_Rotate(Node *x) {
Node *y = x->right;
Node *T2 = y->left;
y->left = x; x->right = T2; // Perform rotation
// Update heights
x->h = Max(Height(x->left), Height(x->right)) + 1;
y->h = Max(Height(y->left), Height(y->right)) + 1;
return y; // Return new root
}
// Get Balance factor of node N
int AVL_Tree :: Get_Balance(Node *N) {
if (N == NULL)
return 0;
return ( Height(N->left) - Height(N->right) );
}
43
Example 1: Implementation of AVL Tree
4

44. void AVL_Tree :: Insert(int key){
root = Insert(root, key);
}
// Recursive function to insert a key in the subtree rooted with node
// and returns the new root of the subtree.
Node* AVL_Tree :: Insert(Node* node, int key) {
/* 1. Perform the normal BST insertion */
if (node == NULL)
return(New_Node(key));
if (key < node->data)
node->left = Insert(node->left, key);
else if (key > node->data)
node->right = Insert(node->right, key);
else // Equal keys are not allowed in BST
return node;
/* 2. Update height of this ancestor node */
node->h = 1 + Max(Height(node->left), Height(node->right));
44
Example 1: Implementation of AVL Tree
5

45. /* 3. Get the balance factor of this ancestor node to check whether
this node became unbalanced */
int b = Get_Balance(node);
// If this node becomes unbalanced, then there are 4 cases
if (b > 1 && key < node->left->data) // Left Left Case
return Right_Rotate(node);
if (b < -1 && key > node->right->data) // Right Right Case
return Left_Rotate(node);
if (b > 1 && key > node->left->data) { // Left Right Case
node->left = Left_Rotate(node->left);
return Right_Rotate(node);
}
if (b < -1 && key < node->right->data) { // Right Left Case
node->right = Right_Rotate(node->right);
return Left_Rotate(node);
}
return node; // return the (unchanged) node pointer
}
45
Example 1: Implementation of AVL Tree
6

46. void AVL_Tree :: Pre_Order( ) {
Pre_Order(root);
}
// A utility function to print preorder traversal of the tree.
void AVL_Tree :: Pre_Order(Node *node) {
if(node != NULL) {
cout << node->data << " ";
Pre_Order(node->left);
Pre_Order(node->right);
}
}
46
Example 1: Implementation of AVL Tree
7

47. int main() {
AVL_Tree atree;
atree.Insert(3); atree.Insert(2);
atree.Insert(1); atree.Pre_Order( ); cout<<endl;
atree.Insert(4); atree.Pre_Order( ); cout<<endl;
atree.Insert(5); atree.Pre_Order( ); cout<<endl;
atree.Insert(6); atree.Pre_Order( ); cout<<endl;
atree.Insert(7); atree.Pre_Order( ); cout<<endl;
atree.Insert(16); atree.Pre_Order( ); cout<<endl;
atree.Insert(15); atree.Pre_Order( ); cout<<endl;
atree.Insert(14); atree.Pre_Order( ); cout<<endl;
atree.Insert(13); atree.Pre_Order( ); cout<<endl;
atree.Insert(12); atree.Pre_Order( ); cout<<endl;
atree.Insert(11); atree.Pre_Order( ); cout<<endl;
atree.Insert(10); atree.Pre_Order( ); cout<<endl;
atree.Insert(8); atree.Pre_Order( ); cout<<endl;
atree.Insert(9); atree.Pre_Order( ); cout<<endl;
system("PAUSE");
return 0;
}
47
Example 1: Implementation of AVL Tree
8