Pattern avoidance in binary trees

TL;DR

This paper develops an algorithm to enumerate binary trees avoiding specific patterns, explores Wilf equivalence among patterns, and proposes a restructuring method to establish bijections between equivalent pattern classes.

Contribution

It introduces a novel algorithm for generating functions of pattern-avoiding binary trees and a general tree restructuring method to prove pattern equivalences.

Findings

Algorithm for counting pattern-avoiding binary trees

Identification of Wilf equivalence classes for tree patterns

A proposed restructuring method for bijective proofs

Abstract

This paper considers the enumeration of trees avoiding a contiguous pattern. We provide an algorithm for computing the generating function that counts n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this counting mechanism, we study the analogue of Wilf equivalence in which two tree patterns are equivalent if the respective n-leaf trees that avoid them are equinumerous. We investigate the equivalence classes combinatorially. Toward establishing bijective proofs of tree pattern equivalence, we develop a general method of restructuring trees that conjecturally succeeds to produce an explicit bijection for each pair of equivalent tree patterns.