  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.
