The independent set sequence of some families of trees

Understand

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.