In this talk I will talk about certain types of particle systems whose interactions are determined by an underlying sparse graph. Such systems often lead to a description of its limiting distribution in terms of a branching distributional equation. Solutions to such equations are usually constructed on weighted branching trees, so the aim of this talk is to explain how these two concepts are related when the underlying sparse graphs are locally tree like, since when this occurs the connection can be explained through an exchange of limits: the limit in time in the particle system and the large graph limit in the underlying graph.