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How can one prove that for every absolute binary tree (every node has either 0 or 2 children) there exists a Huffman series which can be represented by said tree.

Any hints will be much appreciated!

AYR
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1 Answers1

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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).

  1. 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.
  2. It can be proved almost the same way that b_i sequence may produce exactly the given tree if you use a Huffman algorithm.
Ivan Smirnov
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