Enumeration of (uni- or bicolored) plane trees according to their degree distribution

Abstract

The main object of this paper is to give explicit formulas for the number of unlabelled bicolored (unrooted) plane trees with given degree distributions (1 i 1 2 i 2 …; 1 j 1 2 j 2 …). These trees are closely related to the Shabat polynomials, i.e., polynomials over C having at most two critical values (cf. Shabat and Zvonkin, 1994). In the case of planted trees (i.e., rooted at a leaf), this problem was solved by Tutte in 1964, using multivariate Lagrange inversion. Here the key to the solution is the dissymmetry theorem for enriched trees which takes a particularly simple form in the bicolored case and which allows one to get rid of the rooting. We also enumerate those trees in the labelled case, in the unicolored case, as well as when the automorphism group (necessarily cyclic) is of order h equal to (or a multiple of) a given integer k ⩾ 1.