Smoluchowski's coagulation equation is related in remarkable ways to models of random shock clustering and the merging of ancestral trees. I plan to discuss a) work with Iyer and Leger on criteria for the asymptotic self-similarity of clan size distributions in branching processes, and b) how kinetic coagulation models are tied to Menon and Srinivasan's discoveries that indicate complete integrability for Burgers turbulence.