The independent set sequence of some families of trees

For a tree T , let iT (t) be the number of independent sets of size t in T . It is an open question, raised by Alavi, Malde, Schwenk and Erdős, whether the sequence (iT (t))t≥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 Zhu, we show unimodality of the independent set sequence of a path on 2n vertices with `1 and `2 pendant edges dropped alternately from the vertices of the path, `1, `2 arbitrary. Extending another result of Wang and Zhu we show that if T is a tree for which the polynomial ∑ t≥0 iT (t)x t = 0 has only real roots, and if Tk is obtained from T by appending a path of length k at each vertex of T , then (iTk(t))t≥0 is unimodal. We also show that the independent set sequence of any tree becomes unimodal if sufficiently many pendant edges are added to any single vertex. This in particular implies the unimodality of the independent set sequence of some non-periodic caterpillars.