Pattern Avoidance in Ternary Trees

TL;DR

This paper studies the enumeration of ternary trees avoiding specific contiguous patterns, deriving recurrence relations, generating functions, and bijections to understand pattern avoidance and equivalences.

Contribution

It introduces methods for enumerating pattern-avoiding ternary trees, including recurrence relations, generating functions, and bijections, extending known algorithms and concepts from binary trees.

Findings

Derived recurrence relations for simple pattern avoidance

Computed generating functions for complex patterns

Established bijections explaining pattern equivalences

Abstract

This paper considers the enumeration of ternary trees (i.e. rooted ordered trees in which each vertex has 0 or 3 children) avoiding a contiguous ternary tree pattern. We begin by finding recurrence relations for several simple tree patterns; then, for more complex trees, we compute generating functions by extending a known algorithm for pattern-avoiding binary trees. Next, we present an alternate one-dimensional notation for trees which we use to find bijections that explain why certain pairs of tree patterns yield the same avoidance generating function. Finally, we compare our bijections to known "replacement rules" for binary trees and generalize these bijections to a larger class of trees.