We study the geometric decomposition of the (reduced) Persistent Homology Transform (PHT). We focus on two special classes of objects, star shapes and cutouts. We prove that the PHT_0 of the former can be segmented into smaller, simpler regions known as "sectors", and that the PHT of the latter can be decomposed into vines coming from simpler objects. These results provide a solid basis for future implementations of vines' decomposition of PHT, as they considerably reduce the complexity of the problem.