  The tree complex is a simplicial complex defined in recent work of Belk,
Lanier, Margalit, and Winarski with natural applications to mapping class
groups and complex dynamics. In this article, we connect this setting with the
study of certain convex polytopes: associahedra and cyclohedra. Specifically,
we describe a characterization of these polytopes using planar embeddings of
trees and show that the tree complex is the barycentric subdivision of a
polyhedral cell complex for which the cells are products of associahedra and
cyclohedra.
