Define AVL tree, Co nstruct AVL-Treebyinserting a number from 1 to 8

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.

Now, let's construct an AVL tree by inserting numbers from 1 to 8.

1. Start by inserting the first number, 1. This becomes our root node.
2. Insert the next number, 2. Since 2 is greater than 1, it becomes the right child of 1.
3. Insert the next number, 3. Since 3 is greater than both 1 and 2, it would normally become the right child of 2. However, this would unbalance the tree, so a left rotation is performed on 1 to make 2 the new root.
4. Insert the next number, 4. This becomes the right child of 3.
5. Insert the next number, 5. This would normally become the right child of 4, but that would unbalance the tree. So, a left rotation is performed on 2 to make 3 the new root.
6. Insert the next number, 6. This becomes the right child of 5.
7. Insert the next number, 7. This would normally become the right child of 6, but that would unbalance the tree. So, a left rotation is performed on 3 to make 4 the new root.
8. Finally, insert the last number, 8. This becomes the right child of 7.

So, the final AVL tree looks like this:

      4
     / \
    2   6
   / \ / \
  1  3 5  7
            \
             8

This tree is balanced, with each node's child subtrees differing in height by at most one.