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.

翻译：毛毛虫的直径是$\m+m\cdot (H+1)$, 其中一美元是脊椎数, 百万美元是叶子数, 美元是叶子分布的酶。 我们的证据显示毛毛虫与双胞菌之间的紧密联系。 我们用 Wilber 的第一个更低的圈子来测量动态毛毛虫, 通过减少在静态毛片中搜索的问题, 我们证明了更低的界限 。