The minimum height of an AVL tree with n nodes is

The minimum height of an AVL tree with n nodes is calculated based on the property of an AVL tree that it is a balanced binary search tree. This means that for any given node, the heights of its left and right subtrees differ by at most one.

The minimum height h of an AVL tree with n nodes can be calculated using the Fibonacci sequence. The minimum number of nodes in an AVL tree of height h is one more than the sum of the minimum number of nodes in AVL trees of heights h-1 and h-2. This is similar to the Fibonacci sequence, where each number is the sum of the two preceding ones.

So, if we denote the minimum number of nodes in an AVL tree of height h as N(h), we have:

N(h) = N(h-1) + N(h-2) + 1

This is similar to the Fibonacci sequence, but shifted by one place. Therefore, the minimum height of an AVL tree with n nodes is approximately the value h for which N(h) is just less than or equal to n.

This value h can be calculated using the formula for the Fibonacci sequence:

h = floor(log_phi(n)), where log_phi is the logarithm to the base phi (the golden ratio), and floor is the mathematical function that rounds down to the nearest integer.

So, the minimum height of an AVL tree with n nodes is approximately floor(log_phi(n)).