Noncontiguous Pattern Containment in Binary Trees

Alexa Serrato,Lara Pudwell,Tyler Schrock,Connor Scholten

doi:10.1155/2014/316535

Alexa Serrato, Lara Pudwell + Show 2 more

Open Access

https://doi.org/10.1155/2014/316535

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Abstract

We consider the enumeration of binary trees containing noncontiguous binary tree patterns. First, we show that any two ℓ-leaf binary trees are contained in the set of all n-leaf trees the same number of times. We give a functional equation for the multivariate generating function for number of n-leaf trees containing a specified number of copies of any path tree, and we analyze tree patterns with at most 4 leaves. The paper concludes with implications for pattern containment in permutations.

Highlights

Pattern avoidance has been studied in a number of combinatorial objects including permutations, words, partitions, and graphs
In 2012, Dairyko et al [8] considered noncontiguous patterns in binary trees in order to introduce a tree pattern analogue of classical permutation patterns. They showed that for any n, l ∈ Z+, any two l-leaf noncontiguous binary tree patterns are avoided by the same number of n-leaf trees and gave an explicit generating function for this enumeration
Since any two l-leaf tree patterns are a finite sequence of neighboring trees apart, we have that φtp−1,s ∘ ⋅ ⋅ ⋅ ∘ φt2,t3 ∘ φt,t2 provides a bijection between all copies of t in Tn and all copies of s in Tn, so Theorem 3 is true

Summary

Introduction

Pattern avoidance has been studied in a number of combinatorial objects including permutations, words, partitions, and graphs. In 2010, Rowland [6] explored contiguous pattern avoidance in binary trees (i.e., rooted ordered trees in which each vertex has 0 or 2 children). The patterns in [6, 7] may be seen as parallel to consecutive patterns in permutations In those papers, tree T was said to contain tree t as a (contiguous) pattern if t was a contiguous, rooted, ordered, subtree of T. In 2012, Dairyko et al [8] considered noncontiguous patterns in binary trees in order to introduce a tree pattern analogue of classical permutation patterns. They showed that for any n, l ∈ Z+, any two l-leaf noncontiguous binary tree patterns are avoided by the same number of n-leaf trees and gave an explicit generating function for this enumeration.

Definitions and Notation

Total Number of Copies

Pattern Containment of Path Trees

Pattern Containment of Small Trees

Connections to Pattern-Avoiding Permutations

Conflict of Interests