Inspired by the work of Soma and Watari, we define a tree structure on the sub-semimodule of a semi-group Γ associated an irreducible curve singularity (C, O). Building on the results of Oblomkov, Rasmussen and Shende, we show that for certain singularities, this tree encodes some aspects of the geometric structure of the punctual Hilbert schemes of (C, O). As an application, we compute the motivic Hilbert zeta function for some singular curves. A point of the Hilbert scheme corresponds to an ideal in the ring of global sections on (C, O). We study the geometry of subsets of these Hilbert schemes defined by constraints on the minimal number of generators of the defining ideal, and describe some of their geometric properties.
Hajli et al. (Mon,) studied this question.