Generalized Huffman Trees

We generalize the Huffman construction for t-ary trees to the case where the collection of outdegrees for every internal node is given but the outdegrees are not necessarily a constant. The output trees from this construction are called generalized Huffman trees. We prove optimality properties for such trees by generalizing results of Huffman, Hu and Tucker, Glassey and Karp on t-ary trees. We also give a criterion to compare the costs of two generalized Huffman trees with the same collection of outdegrees. This criterion, when applied to binary trees, considerably strengthens a result of Hu.