Variations on a theme by Huffman

Abstruct-In honor of the twenty-fifth amdversary of Huffman coding, four new results about Huffman codes are presented. 'Ibe ffrst result shows that a binary prefix condition code is a Huffmau code iff the intermediate and terminal nodes id the code tree can be listed by nonin-creasing probabiuty so that each node in the fist is adjacent to its sibling. The second result upper bounds the redundancy (expected length minus entropy) of a bii Huffman code by P, + log2[2(log2e)/e] = P, + 0.086, where P, is the probability of the most likely source letter. 'zhe third result shows that one can always leave a codeword of length two unused and stffl have a redundancy of at most one. The fourth result is a simple algorithm for adapting a Huffmau code to slowly vary@ e&mates of the source probabilities. In essence, one maintains a running count of uses of each node in the code tree and lists the nodes in order of these counts. Whenever the occurrence of a message increases a node count above the