What is the depth of a complete binary tree with $N$ nodes?

Question Detail: 

This question uses the following definition of a complete binary tree^†:

A binary tree $T$ with $N$ levels is complete if all levels except possibly the last are completely full, and the last level has all its nodes to the left side.

The following is an excerpt from Algorithms:

It ($\log N$) is also the depth of a complete binary tree with $N$ nodes. (More precisely: $⌊\log N⌋$.)

Why is the above excerpt true?

^† Originally defined here

Asked By : Farhad Yusufali

Best Answer from StackOverflow

Question Source : http://cs.stackexchange.com/questions/6161

Answered By : Luke Mathieson

Consider how a complete binary tree of height $h$ is constructed, one vertex at the root level, two at the first level below the root, four at the second level below, and so on, until the $h^{th}$ level, which has at least one vertex, but at most twice as many as the previous level. Note that the number of vertices at each level is a power of two (excluding the last, which is a special case). Then we have: $$ 1+\sum_{i=0}^{h-1}2^{i} \leq n \leq \sum_{i=0}^{h}2^{i} $$ Using the identity that the sum of the first $k$ powers of two is $2^{k+1}-1$ we get: $$ 1+2^{h}-1 \leq n \leq 2^{h+1}-1\\ 2^{h} \leq n \leq 2^{h+1}-1 $$ and hence $$ 2^{h} \leq n < 2^{h+1} $$

Taking the base 2 logarithm: $$ h \leq \log n < h+1 $$ So we can conclude $$h = \lfloor\log n\rfloor$$ As $\log n$ is bigger than $h$, but less than the next integer $h+1$.