Fast reversible coding and indexing of colorations

Given a non-branched tree withn vertices. Then, by ann-g-coloration we understand a partition of the set of vertices into no more thang classes, such that adjacent vertices belong to different classes. Supposed the set (mathfrak{S}) of alln-g-colorations (for givenn andg) is lexicographically ordered, here are given two algorithms: the first directly determines (without using the set proper) the ordinal number of an arbitrary element of (mathfrak{S}) ; the other directly generates an element of (mathfrak{S}) from its given ordinal number.