Intervals of balanced binary trees in the Tamari lattice

TL;DR

This paper explores the structure of balanced binary trees within the Tamari lattice, showing their interval properties, isomorphisms to hypercubes, and introducing new grammatical and pattern-based methods for their enumeration and analysis.

Contribution

It demonstrates that balanced binary trees form intervals in the Tamari lattice, establishes their isomorphism to hypercubes, and introduces new grammatical and pattern tools for their enumeration.

Findings

Intervals of balanced binary trees are isomorphic to hypercubes.

Synchronous grammars can generate and enumerate these trees.

Imbalance tree patterns describe special subsets of balanced trees.

Abstract

We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T, T'] where T and T' are balanced binary trees are isomorphic as posets to a hypercube. We introduce synchronous grammars that allow to generate tree-like structures and obtain fixed-point functional equations to enumerate these. We also introduce imbalance tree patterns and show that they can be used to describe some sets of balanced binary trees that play a particular role in the Tamari lattice. Finally, we investigate other families of binary trees that are also closed by interval in the Tamari lattice.