1. What is AVL tree?

AVL tree is a self-balancing binary search tree that was invented by Adelson-Velsky and Landis in 1962. It is named after its inventors and is widely used in computer science for efficient searching, insertion, and deletion operations. The key feature of AVL tree is that it maintains a balance between its left and right subtrees, ensuring that the height difference between them is at most 1.

2. History of AVL tree

AVL tree was invented by Adelson-Velsky and Landis in 1962, and it was the first data structure to be invented that guaranteed worst-case logarithmic time complexity for searching, insertion, and deletion operations. The name “AVL” stands for Adelson-Velsky and Landis, and it was chosen to honor the inventors of this data structure.

3. Properties of AVL tree

AVL tree has several important properties that make it an efficient data structure for storing sorted data. Some of these properties are:

3.1 Height-balanced property

AVL tree maintains a balance between its left and right subtrees, ensuring that the height difference between them is at most 1. This property is crucial for ensuring that the tree remains balanced even after insertion or deletion operations.

3.2 Balance factor

The balance factor of a node in AVL tree is the difference between the heights of its left and right subtrees. If the balance factor of a node is greater than 1 or less than -1, then the tree is unbalanced and needs to be rebalanced.

3.3 Rotations

AVL tree uses four types of rotations to rebalance itself: LL rotation, RR rotation, LR rotation, and RL rotation. These rotations are performed to maintain the height-balanced property of the tree.

4. Insertion in AVL tree

When a new node is inserted into an AVL tree, the tree may become unbalanced. To restore the balance, one of the four rotations is performed based on the balance factor of the node.

4.1 Case 1: LL Rotation

If the balance factor of the node is greater than 1 and the balance factor of its left child is greater than or equal to 0, then a LL rotation is performed.

4.2 Case 2:RR Rotation

If the balance factor of the node is less than -1 and the balance factor of its right child is less than or equal to 0, then a RR rotation is performed.

4.3 Case 3: LR Rotation

If the balance factor of the node is greater than 1 and the balance factor of its left child is less than 0, then a LR rotation is performed.

4.4 Case 4: RL Rotation

If the balance factor of the node is less than -1 and the balance factor of its right child is greater than 0, then a RL rotation is performed.

5. Deletion in AVL tree

When a node is deleted from an AVL tree, the tree may become unbalanced. To restore the balance, one of the four rotations is performed based on the balance factor of the node.

5.1 Case 1: Deleting a leaf node

If the node to be deleted is a leaf node, it is simply removed from the tree.

5.2 Case 2: Deleting a node with one child

If the node to be deleted has only one child, the child replaces the node.

5.3 Case 3: Deleting a node with two children

If the node to be deleted has two children, the node is replaced by its successor or predecessor, and the successor or predecessor is deleted.

6. Time complexity of AVL tree operations

The time complexity of AVL tree operations depends on the height of the tree. Since AVL tree maintains a balance between its left and right subtrees, the height of the tree is always logarithmic to the number of nodes in the tree. Therefore, the time complexity of AVL tree operations such as searching, insertion, and deletion is O(log n), where n is the number of nodes in the tree.

7. Comparison with other balanced search trees

AVL tree is one of the many balanced search trees that are used in computer science. Some of the other balanced search trees are Red-Black tree, B-tree, and Splay tree. AVL tree is more balanced than Red-Black tree, which means that AVL tree has a smaller height than Red-Black tree for the same number of nodes. However, AVL tree is less efficient than B-tree and Splay tree in some cases.

8. Applications of AVL tree

AVL tree is widely used in computer science for storing sorted data and performing efficient searching, insertion, and deletion operations. Some of the applications of AVL tree are:

  • Databases
  • Indexing
  • Compilers
  • Language parsing
  • Graph algorithms

9. Advantages and disadvantages of AVL tree

AVL tree has several advantages and disadvantages, some of which are:

9.1 Advantages

  • Efficient searching, insertion, and deletion operations
  • Guaranteed worst-case logarithmic time complexity
  • Self-balancing

9.2 Disadvantages

  • Higher memory usage than other data structures
  • Slower insertion and deletion operations than other data structures
  • Complex implementation