Huffman coding with an infinite alphabet

A new type of sufficient condition is provided for a probability distribution on the nonnegative integers to be given an optimal D-ary prefix code by a Huffman-type algorithm. In the justification of our algorithm, we introduce two new (essentially one) concepts as the definition of the "optimality" of a prefix D-ary code, which are shown to be equivalent to that defined in the traditional way. These new concepts of the optimality are meaningful even for the case where the Shannon entropy H(P) diverges.