Lexicographic Listing and Ranking of r-ary Trees

This paper presents three simple and efficient algorithms for generating, ranking and unranking t-ary trees in a lexicographic order. The simplest idea of encoding a t-ary tree with n nodes as a bit-string of length t*n is exploited to its full advantages. It is proved that the lexicographic order in the set of t-ary trees with n nodes is preserved in the set of bit-strings of length t*n, using the above encoding scheme. Thus by generating all bit-strings in the lexicographic order, a simple decoding algorithm can convert them to t-ary trees in the same order. Finally, the theoretical basis for ranking a lexicographic listing of bit-strings is discussed, and the ranking and the unranking algorithms are derived.