The diameter of caterpillar associahedra

The caterpillar associahedron $\mathcal{A}(G)$ is a polytope arising from the rotation graph of search trees on a caterpillar tree $G$, generalizing the rotation graph of binary search trees (BSTs) and thus the conventional associahedron. We show that the diameter of $\mathcal{A}(G)$ is $\Theta(n + m \cdot (H+1))$, where $n$ is the number of vertices, $m$ is the number of leaves, and $H$ is the entropy of the leaf distribution of $G$. Our proofs reveal a strong connection between caterpillar associahedra and searching in BSTs. We prove the lower bound using Wilber's first lower bound for dynamic BSTs, and the upper bound by reducing the problem to searching in static BSTs.