Problems on Deligne-Mumford spaces

poset underlying associahedra

As described in Deligne-Mumford space, for any $d\geq 1$, $\overline {\mathcal {M}}_{{d+1}}$ is a $(d-2)$-dimensional manifold with boundary and corners which parametrizes nodal trees of disks with $d+1$ marked points, one of them distinguished (we think of the $d$ undistinguished resp. 1 distinguished marked points as "input" resp. "output" marked points). As shown in [Auroux, Ex. 2.6], $\overline {\mathcal {M}}_{4}$ is homeomorphic to a closed interval, with one endpoint corresponding to a collision of the first two inputs $z_{1},z_{2}$ and the other corresponding to a collision of $z_{2},z_{3}$. Work out which polygon/polyhedron $\overline {\mathcal {M}}_{5},\overline {\mathcal {M}}_{6}$ are equal to. (Keep in mind that when $\geq 3$ marked points collide simultaneously, there is a continuous family