霍夫曼树的主张？

Question

I saw the following claim:

Given Q={1,2,...,n} , and positive frequency function f such that:

f(1) > f(2) >... > f(n) > f(1)/3 ,

Then there are leafs at maximum 3 different levels of Huffman tree.

I have been looking for a counterexample but no luck, can someone help me?

Answer 1

Assume that the claim is false for some tree, with shallowest leaf at depth H. The shallowest leaf will have frequency f(1).

Since the tree extends to depth at least H+3, then there must be a subtree at depth H+1 with at least 3 leaves. The smallest possible subtree that gets to H+3 is shaped like:

   O
  / \
 O   O
    / \
   O   O

The total frequency of the subtree is more than f(1), the frequency of the shallowest leaf, and yet it occurs at a deeper level. We could therefore improve the Huffman tree by swapping the positions of this subtree and the shallowest leaf.

Since Huffman trees are proven to be optimal, this can't happen, so the claim must be true.