On the Diameter of Tree Associahedra

TL;DR

This paper establishes that the worst-case diameter of tree associahedra, representing search tree transformations on trees, is on the order of n log n, providing a tight bound and answering a previously open question.

Contribution

The authors construct specific tree families demonstrating an Omega(n log n) diameter, proving the tight bound for the worst-case diameter of tree associahedra.

Findings

Worst-case diameter of tree associahedra is Theta(n log n)

Constructed tree families with Omega(n log n) transformation steps

Introduced a projection method for search trees

Abstract

We consider a natural notion of search trees on graphs, which we show is ubiquitous in various areas of discrete mathematics and computer science. Search trees on graphs can be modified by local operations called rotations, which generalize rotations in binary search trees. The rotation graph of search trees on a graph $G$ is the skeleton of a polytope called the graph associahedron of $G$. We consider the case where the graph $G$ is a tree. We construct a family of trees $G$ on $n$ vertices and pairs of search trees on $G$ such that the minimum number of rotations required to transform one search tree into the other is $\Omega (n\log n)$. This implies that the worst-case diameter of tree associahedra is $\Theta (n\log n)$, which answers a question from Thibault Manneville and Vincent Pilaud. The proof relies on a notion of projection of a search tree which may be of independent interest.