Counting Colored Trees

TL;DR

This paper studies the enumeration of colored plane trees under specific coloring rules, establishing conditions for sequence equivalence, and deriving formulas for sequences including Fibonacci, Catalan, Narayana, and Schr"oder.

Contribution

It provides a comprehensive analysis of counting sequences for colored trees, including conditions for equivalence and explicit formulas for various coloring schemes.

Findings

Identified conditions for different coloring rules to produce the same counting sequence.

Derived formulas and closed-form expressions for sequences like Fibonacci, Catalan, Narayana, and Schr"oder.

Extended coloring rules to arbitrary numbers of colors, broadening the scope of enumeration.

Abstract

We consider the enumeration of plane trees (rooted ordered trees) whose vertices are colored according to a specific coloring rule that prescribes which possible pairs of colors can occur as the colors of a parent vertex and its child. This general construction covers many different examples that have been studied in the literature. Some general necessary and sufficient conditions for two different coloring rules to result in the same counting sequence are established. We also provide exhaustive lists of counting sequences arising from coloring rules with two or three colors, and we find formulas and closed form expressions for many of these sequences. The famous Fibonacci, Catalan, Narayana, and Schr\"oder sequences appear in several cases. Some of these coloring rules are extended to families of coloring rules with arbitrarily many colors.