In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters tend to require local finiteness, and hence finite generation of the group. In this paper, I take the theory of intersections of splittings of finitely generated groups (as developed by Scott, Scott-Swarup, and Niblo-Sageev-Scott-Swarup), and rework it to remove finite generation assumptions. Whereas the aforementioned authors relied on the local finiteness of the Cayley graph, I capitalize on the Bass-Serre trees for the splittings.