We will now proceed to prove a number of theorems about perfect binary trees.

**1) Nodes in a Perfect N-ary Tree : Theorem**

**2) Logarithmic Height**

and therefore logN(n) is a reasonable approximation of the height of a perfect N-ary tree with n nodes.

**3) N-ary Trees versus Binary Trees**

The ratio of the heights of a binary tree containing n nodes and an N-ary tree containing n nodes is

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