Balanced binary trees in the Tamari lattice

TL;DR

This paper demonstrates that balanced binary trees form intervals in the Tamari lattice, with these intervals being isomorphic to hypercubes, and introduces new methods for enumerating such intervals.

Contribution

It establishes the closure of balanced binary trees under intervals in the Tamari lattice and introduces tree patterns and synchronous grammars for enumeration.

Findings

Intervals of balanced trees are isomorphic to hypercubes

Balanced trees form closed intervals in the Tamari lattice

New enumeration methods using tree patterns and grammars

Abstract

We show that the set of balanced binary trees is closed by interval in the Tamari lattice. We establish that the intervals [T0, T1] where T0 and T1 are balanced trees are isomorphic as posets to a hypercube. We introduce tree patterns and synchronous grammars to get a functional equation of the generating series enumerating balanced tree intervals.