Inference problems on gene-orders have suffered limited applicability due to complex models, and complications related to preprocessing the data. For instance, we lack practical ways to sample rearrangement scenarios, and traditional estimates of breakpoint re-use are highly dependent on the resolution of the syntenic blocks. We first simplify the task of counting scenarios by relating genome rearrangement scenarios to quadragulations and planar trees of a convex polygon. Next, we present an optimization problem for computing breakpoint re-use on a phylogeny, rather than between a single pair of genomes.