What is the number of nodes at a particular level in a balanced binary search tree?

Question

I was asked this question in a phone screen interview and I was not able to answer it. For example, in a BST, I know that the maximum number of nodes is given by 2^h (assuming the root node at height = 0)

I wanted to ask, is there a similar mathematical outcome for a balanced binary search tree as well (For AVL, Red Black trees?), ie the number of nodes at a particular level k.

Thanks!

Answer 1

A balanced binary tree starts with one node, which has two descendants. Each of those then has two descendants again. So there will be 1, 2, 4, 8 and so on nodes per level.

As a formula you can use 2^(level-1) . The last row might not be completely full, so it can have less elements.

As the balancing step is costly, implementations usually do not rebalance after every mutation of the tree. They will rather apply a heuristic to find out when a rebalancing will make the most sense. So in practice, levels might have less nodes than if the tree were perfectly balanced and there might be additional levels from nodes being inserted in the wrong places.