  Alavi, Malde, Schwenk and Erd\H{o}s asked whether the independent set
sequence of every tree is unimodal. Here we make some observations about this
question. We show that for the uniformly random (labelled) tree, asymptotically
almost surely (a.a.s.) the initial approximately 49.5\% of the sequence is
increasing while the terminal approximately 38.8\% is decreasing. Our approach
uses the Matrix Tree Theorem, combined with computation. We also present a
generalization of a result of Levit and Mandrescu, concerning the final
one-third of the independent set sequence of a K\"onig-Egerv\'ary graph.
