The tree complex is a simplicial complex which was recently introduced in recent work on complex dynamics by Belk, Lanier, Margalit and Winarski. Each simplex in the complex is defined by a certain type of embedding for trees in the plane, and incidence is determined by a particular type of edge-contraction. In this talk I will describe a new polyhedral structure for the tree complex, in which each cell is a product of associahedra and cyclohedra, along with some connections to the combinatorics of noncrossing partitions and the braid group.