Self-Reachable Configuration Polytopes for Trees

We study lattice polytopes which arise as the convex hull of chip vectors for $\mathit{\text{self-reachable}}$ chip configurations on a tree $T$. We show that these polytopes always have the integer decomposition property and characterize the vertex sets of these polytopes. Additionally, in the case of self-reachable configurations with the smallest possible number of chips, we show that these polytopes are unimodularly equivalent to a unit cube.