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Counting Colored Trees

Stoyan Dimitrov, Nathan Fox, Kimberly Hadaway, Ashley Tharp, Stephan Wagner

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 Schröder sequences appear in several cases. Some of these coloring rules are extended to families of coloring rules with arbitrarily many colors.

Counting Colored Trees

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 Schröder sequences appear in several cases. Some of these coloring rules are extended to families of coloring rules with arbitrarily many colors.
Paper Structure (23 sections, 41 theorems, 149 equations, 4 figures, 1 table)

This paper contains 23 sections, 41 theorems, 149 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Other related work and summary of the paper
  3. Preliminaries
  4. Tree coloring equivalence
  5. Necessary conditions
  6. Sufficient conditions
  7. Coloring rules based on block matrices
  8. Small matrices representing coloring rules
  9. Coloring rules with two colors
  10. Coloring rules with three colors
  11. Equivalent matrices that are not strongly equivalent
  12. An appearance of Fibonacci numbers
  13. Sequences related to Entry 7 in Table \ref{['tb:2cols']}
  14. Special families of coloring rules
  15. Family 1
  16. ...and 8 more sections

Key Result

Theorem 1

The total number of independent sets over all plane trees with $n$ vertices is

Figures (4)

  • Figure 1: All $4$-vertex plane trees.
  • Figure 2: All $3$-vertex plane trees with leaves that can either be blue or white (while all other vertices are blue).
  • Figure 3: The standard Catalan decomposition for plane trees
  • Figure 4: Example of the glove bijection for plane trees.

Theorems & Definitions (74)

  • Theorem 1: see KPT1986Fibonacci
  • Theorem 2
  • Theorem 3: see GPW2010Bijections
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • Theorem 6
  • proof
  • Corollary 1
  • ...and 64 more