Stirling Decomposition of Graph Homology in Genus 1

We prove that commutative graph homology in genus $g=1$ with $n\ge 3$ markings has a direct sum decomposition whose summands have rank given by Stirling numbers of the first kind. These summands are computed as the homology of complexes of certain decorated trees. This paper was written with a non-expert audience in mind, and an emphasis is placed on an elementary combinatorial description of these decorated tree complexes.