# Two definitions of balanced binary trees

I have seen two definitions of balanced binary trees, which look different to me.

1. A binary tree is balanced if for each node it holds that the number of inner nodes in the left subtree and the number of inner nodes in the right subtree differ by at most 1.

2. A binary tree is balanced if for any two leaves the difference of the depth is at most 1.

Does every tree that satisfies def. 1 also satisfy def. 2? What about the other way round?

• Have you tried proving either direction? What are your findings? – Raphael Sep 12 '12 at 22:31

2. In other words, such a tree of height $k$ without the leaves at level $k$ is a perfect tree of height $k-1$.