(a) Define AVL Tree. Discuss various types of nodes required to implement an AVL tree. (b) What is an unbalanced AVL tree? Construct an AVL tree for the following: March, May, November, August, April, June, December, July, February, June, October, September

Q.) (a) Define AVL Tree. Discuss various types of nodes required to implement an AVL tree. (b) What is an unbalanced AVL tree? Construct an AVL tree for the following: March, May, November, August, April, June, December, July, February, June, October, September
Subject: Data Structure and Algorithm

(a) Define AVL Tree

An AVL tree is a self-balancing binary search tree, where the difference between heights of left and right subtrees cannot be more than one for all nodes. This property of AVL trees ensures that the height of the tree remains O(log n) where n is the number of nodes in the tree, thus maintaining the operations such as insertion, deletion, and look-up to be performed in logarithmic time.

Types of Nodes Required to Implement an AVL Tree

To implement an AVL tree, we need to define a structure for nodes that will include the following:

Key: The value or data that is stored in the node.
Left Child: A pointer to the left child node.
Right Child: A pointer to the right child node.
Height: The height of the node in the tree. The height of a leaf node is considered to be 0.

Here is a simple representation of an AVL tree node:

struct AVLNode {
    int key;
    struct AVLNode *left;
    struct AVLNode *right;
    int height;
};

(b) What is an Unbalanced AVL Tree?

An unbalanced AVL tree is a binary search tree that does not satisfy the AVL property, i.e., there exists at least one node in the tree where the difference in heights between the left and right subtrees is greater than one. This situation arises after operations like insertion or deletion and requires rebalancing to restore the AVL property.

Constructing an AVL Tree

To construct an AVL tree from the given list of months, we will insert the nodes one by one and balance the tree if necessary. The months can be represented by their lexicographical order, for example, April (4), August (8), December (12), February (2), July (7), June (6), March (3), May (5), November (11), October (10), September (9).

Let's construct the AVL tree step by step:

Insert "March" (3)
```
3
/ \
null null
```
Insert "May" (5)
```
3
/ \
null  5
```
Insert "November" (11)
```
3
/ \
null  5
   \
   11
```
After inserting "November", the tree becomes unbalanced. We perform a right rotation on the root:

    5
   / \
  3  11

Insert "August" (8)
```
5
/ \
3  11
 /
8
```
Insert "April" (4)
```
5
/ \
3  11
/  /
4  8
```
After inserting "April", the tree becomes unbalanced. We perform a left-right rotation on the subtree rooted at 3:

Insert "June" (6) 5 / \ 4 11 / / \ 3 8 6 After inserting "June", the tree becomes unbalanced. We perform a right-left rotation on the subtree rooted at 11:

Insert "December" (12)

Insert "July" (7)

Insert "February" (2)

Insert "June" (6) again Since "June" (6) is already in the tree, we ignore this duplicate entry.

Insert "October" (10)

Insert "September" (9)
```
5
/ \
4   8
/   / \
2   6  11
/ \   \
3  7   12
     /
   10
  /
 9
```
After inserting "September", the tree becomes unbalanced. We perform a left rotation on the subtree rooted at 11:

The final AVL tree is balanced with the AVL property maintained after each insertion. Note that rotations are crucial operations to maintain the balance of the tree. There are four types of rotations: left rotation, right rotation, left-right rotation, and right-left rotation. These rotations are used to rebalance the tree whenever the AVL property is violated.