Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 31 / 1 / b

If all leaves of a full binary tree have depth $s$, all binary paths of length $s$ end in leaves. Thus it has exactly $2^s$ leaves and $m=2^s$. Equivalently, equality in the Kraft inequality gives $m{2}^{-s}=1$.

Conversely, when $m=2^k$ a complete depth-$k$ tree is a prefix code with mean length $k$. For any prefix lengths $s_i$, convexity givesso $L\ge {\log }_{2}m=k$. The complete tree is optimal. A Huffman code attains this optimum, and equality in the strictly convex first inequality forces all its lengths to equal $k$. Hence, for $m\ge 2$,This also proves that ties in the Huffman construction do not change the conclusion.