Explain AVL tree. Insert the following elements in AVL search tree 48, 25, 10, 5, 7, 3, 28, 8, 16

Q.) Explain AVL tree. Insert the following elements in AVL search tree 48, 25, 10, 5, 7, 3, 28, 8, 16
Subject: Data Structures - II

An AVL tree (named after inventors Adelson-Velsky and Landis) is a self-balancing binary search tree. In an AVL tree, the heights of the two child subtrees of any node differ by at most one; if at any time they differ by more than one, rebalancing is done to restore this property. Lookup, insertion, and deletion all take O(log n) time in both the average and worst cases, where n is the number of nodes in the tree prior to the operation. Insertions and deletions may require the tree to be rebalanced by one or more tree rotations.

The balance factor of a node is the height of its left subtree minus the height of its right subtree and a node with balance factor 1, 0, or -1 is considered balanced. A node with any other balance factor is considered unbalanced and requires rebalancing the tree. The balance factor is either stored directly at each node or computed from the stored heights of the subtrees.

Now, let's insert the given elements into an AVL tree.

Start by inserting the first element 48.
Insert the next element 25. Since it's less than 48, it goes to the left of 48. The tree is still balanced.
Insert 10. It goes to the left of 25. Now, the tree becomes unbalanced. The balance factor of the root node 48 becomes 2 which is not allowed. So, perform a right rotation on 48.
Insert 5. It goes to the left of 10. The tree is still balanced.
Insert 7. It goes to the right of 5. Now, the tree becomes unbalanced. The balance factor of the node 10 becomes 2 which is not allowed. So, perform a left rotation on 10.
Insert 3. It goes to the left of 5. The tree is still balanced.
Insert 28. It goes to the right of 25. The tree is still balanced.
Insert 8. It goes to the right of 7. Now, the tree becomes unbalanced. The balance factor of the node 5 becomes -2 which is not allowed. So, perform a right rotation on 5 and then a left rotation on 10.
Insert 16. It goes to the left of 25. The tree is still balanced.

The final AVL tree after inserting all elements would look like this:

        10
       /  \
      7    25
     / \   /  \
    5   8 16   48
   /       /  
  3       28

This is a simplified explanation and the actual process involves more steps including updating the heights of the nodes and checking the balance factor after each insertion.