It has been discovered that the worst-case performance of BST is similar to that of linear search algorithms, namely O (n). We cannot forecast data patterns and frequencies in real-time data. As a result, there is a need to balance out the present BST.
AVL trees are height-balancing binary search trees named after their inventors Adelson, Velski, and Landis. The AVL tree compares the heights of the left and right sub-trees and ensures that the difference is less than one. This distinction is known as the Balance Factor.
Here we see that the first tree is balanced and the next two trees are not balanced −
The left subtree of C in the second tree has a height of 2 and the right subtree has a height of 0, hence the difference is 2. The difference is 2 again in the third tree, since the right subtree of A has height 2 and the left is missing, therefore it is 0. The AVL tree allows only one difference (balancing factor).
BalanceFactor = height(left-subtree) − height(right-subtree)
AVL Rotations
To balance itself, an AVL tree may perform the following four kinds of rotations −
- Left rotation
- Right rotation
- Left-Right rotation
- Right-Left rotation
Left 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 −
In our example, node A has become unbalanced as a node is inserted in the right subtree of A’s right subtree. We perform the left rotation by making A the left-subtree of B.
Right 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.
As depicted, the unbalanced node becomes the right child of its left child by performing a right rotation.
Left-Right Rotation
Double rotations are slightly complex version of already explained versions of rotations. To understand them better, we should take note of each action performed while rotation. Let’s first check how to perform Left-Right rotation. A left-right rotation is a combination of left rotation followed by right rotation.
| State | Action |
|---|---|
 | A node has been inserted into the right subtree of the left subtree. This makes C an unbalanced node. These scenarios cause AVL tree to perform left-right rotation. |
 | We first perform the left rotation on the left subtree of C. This makes A, the left subtree of B. |
 | Node C is still unbalanced, however now, it is because of the left-subtree of the left-subtree. |
 | We shall now right-rotate the tree, making B the new root node of this subtree. C now becomes the right subtree of its own left subtree. |
 | The tree is now balanced. |
Right-Left Rotation
The second type of double rotation is Right-Left Rotation. It is a combination of right rotation followed by left rotation.
| State | Action |
|---|---|
 | A node has been inserted into the left subtree of the right subtree. This makes A, an unbalanced node with balance factor 2. |
 | First, we perform the right rotation along C node, making C the right subtree of its own left subtree B. Now, B becomes the right subtree of A. |
 | Node A is still unbalanced because of the right subtree of its right subtree and requires a left rotation. |
 | A left rotation is performed by making B the new root node of the subtree. A becomes the left subtree of its right subtree B. |
| The tree is now balanced. |
