The independent set sequence of some families of trees

For a tree $T$, let $i_T(t)$ be the number of independent sets of size $t$ in $T$. It is an open question, raised by Alavi, Malde, Schwenk and Erd\H{o}s, whether the sequence $(i_T(t))_{t \geq 0}$ is always unimodal. Here we answer the question in the affirmative for some recursively defined families of trees, specifically paths with auxiliary trees dropped from the vertices in a periodic manner. In particular, extending a result of Wang and B.-X. Zhu, we show unimodality of the independent set sequence of a path on $2n$ vertices with $\ell_1$ and $\ell_2$ pendant edges dropped alternately from the vertices of the path, $\ell_1, \ell_2$ arbitrary. We also show that the independent set sequence of any tree becomes unimodal if sufficiently many pendant edges are dropped from any single vertex, or if $k$ pendant edges are dropped from every vertex, for sufficiently large $k$. This in particular implies the unimodality of the independent set sequence of some non-periodic caterpillars.