Depth and Stanley depth of powers of the edge ideals of some caterpillar and lobster trees

Let $S$ be a ring of polynomials in finitely many variables over a field. In this paper we give lower bounds for depth and Stanley depth of modules of the type $S\mathrm{/}{I}^t$ for $t\ge 1$, where $I$ is the edge ideal of some caterpillar and lobster trees. These new bounds are much sharper than the existing bounds for the classes of ideals we considered.