Finding the minimum and maximum height in a AVL tree, given a number of nodes?

Question

Is there a formula to calculate what the maximum and minimum height for an AVL tree, given a certain number of nodes?

For example:
Textbook question

Answer 1

It's important to note the following defining characteristics of an AVL Tree.

AVL Tree Property

• The nodes of an AVL tree abide by the BST property
• AND The heights of the left and right sub-trees of any node differ by no more than 1.

Theorem: The AVL property is sufficient to maintain a worst case tree height of O(log N).

Note the following diagram.

- T1 is comprised of a T0 + 1 node, for a height of 1.
- T2 is comprised of T1 and a T0 + 1 node, giving a height of 2.
- T3 is comprised of a T2 for the left sub-tree and a T1 for the right sub-tree + 1 node, for a height of 3.
- T4 is comprised of a T3 for the left sub-tree and a T2 for the right sub-tree + 1 node, for a height of 4.

If you take the ceiling of O(log N), where N represents the number of nodes in an AVL tree, you get the height.

Example) T4 contains 12 nodes. [ceiling]O(log 12) = 4.

See the pattern developing here??

**The worst-case height is