Combinatorial generation via permutation languages. VI. Binary trees

TL;DR

The paper extends pattern avoidance in binary trees to a mixed framework where edges carry contiguity labels, establishing a bijection with mesh-pattern avoidance in 231-avoiding permutations. This enables efficient generation of pattern-avoiding trees using the Hartung–Hoang–Mütze–Williams framework via simple greedy slides (Algorithm S) and a history-free variant (Algorithm H), under friendly pattern conditions. It also proves a general tree-to-mesh correspondence, enumerates patterns up to five vertices, and builds rich bijections to bitstrings, Motzkin paths (including catastrophes and colored steps), and non-crossing set partitions, shedding light on Wilf-equivalences among tree patterns. The work advances algorithmic generation and structural understanding of Catalan-type objects through permutation-encoded pattern avoidance, with several open problems and conjectures opening directions for future research and applications in related combinatorial families.

Abstract

In this paper we propose a notion of pattern avoidance in binary trees that generalizes the avoidance of contiguous tree patterns studied by Rowland and non-contiguous tree patterns studied by Dairyko, Pudwell, Tyner, and Wynn. Specifically, we propose algorithms for generating different classes of binary trees that are characterized by avoiding one or more of these generalized patterns. This is achieved by applying the recent Hartung-Hoang-Mütze-Williams generation framework, by encoding binary trees via permutations. In particular, we establish a one-to-one correspondence between tree patterns and certain mesh permutation patterns. We also conduct a systematic investigation of all tree patterns on at most 5 vertices, and we establish bijections between pattern-avoiding binary trees and other combinatorial objects, in particular pattern-avoiding lattice paths and set partitions.

Figures (31)

• Figure 1: Illustration of different notions of pattern containment in binary trees. Contiguous edges are drawn solid, whereas non-contiguous edges are drawn dotted.
• Figure 2: Rotation in binary trees.
• Figure 3: The Lucas--Roelants van Baronaigien--Ruskey algorithm to generate all binary trees with $n=4$ vertices by tree rotations. The vertices are labeled with $1,2,3,4$ according to the search tree property.
• Figure 4: Illustration of definitions related to binary trees.
• Figure 5: Illustration of our notion of pattern containment in binary trees.

Theorems & Definitions (39)

• Lemma 1
• Theorem 2
• proof
• Theorem 3
• Theorem 4: MR4391718
• Lemma 5: MR4391718
• Lemma 6: MR4391718
• Lemma 7: MR4391718
• Lemma 8
• proof