Non-Contiguous Pattern Avoidance in Binary Trees

TL;DR

This paper studies the enumeration of binary trees avoiding specific non-contiguous patterns, providing formulas, generating functions, and connections to permutation pattern avoidance, advancing understanding of pattern-avoiding structures.

Contribution

It introduces explicit formulas and generating functions for counting non-contiguous pattern-avoiding binary trees, and explores their relation to permutation pattern avoidance.

Findings

Closed formulas for trees avoiding single patterns with ≤4 leaves

Explicit generating functions for non-contiguous pattern avoidance

Connections established between pattern-avoiding trees and permutations

Abstract

In this paper we consider the enumeration of binary trees avoiding non-contiguous binary tree patterns. We begin by computing closed formulas for the number of trees avoiding a single binary tree pattern with 4 or fewer leaves and compare these results to analogous work for contiguous tree patterns. Next, we give an explicit generating function that counts binary trees avoiding a single non-contiguous tree pattern according to number of leaves. In addition, we enumerate binary trees that simultaneously avoid more than one tree pattern. Finally, we explore connections between pattern-avoiding trees and pattern-avoiding permutations.