Constructive proof: if a_1, ..., a_k are the leaves' depths, then the series may be b_1 = 2^(-a_1), ..., b_k = 2^(-a_k).
- Sum of all b_i = 1. Proof by induction: for one node this is true; two sibling leaves of depth A may be reduced to their parent and will give one node of depth A-1, while the sum of b_i will not change.
- It can be proved almost the same way that b_i sequence may produce exactly the given tree if you use a Huffman algorithm.