Unimodality of Forest Independence Polynomials

Abstract

In 1987, Alavi, Malde, Schwenk and Erdős asked whether the independence sequence of every tree, and more generally every forest, is unimodal. This long-standing problem asks whether acyclicity alone controls the shape of these counting coefficients, despite unrestricted degrees and branching. Even trees can fail to be log-concave, ruling out that stronger property as a general route to unimodality. We prove the conjecture for all finite forests. Our computer-assisted proof combines structural counting inequalities with hard-core conditional-binomial reductions and exact integer and rational certificates. We reduce the large-order threshold to 59 vertices and extend the finite-order coverage through 60 vertices; the overlapping ranges complete the proof. OpenAI's GPT 6.0 Astra assisted with the computational work; the authors provided the overall proof strategy. This record archives the PDF generated by arXiv and the unchanged LaTeX source package submitted on 27 September 2026. Verification code and certificate data are available at the fixed repository snapshot cited in the paper: https://github.com/zhangzenozhang-jpg/forest-unimodality-arxiv-verification/tree/806789352f7ed8e64594907a2ed334652ce11160/companion/2026-09-27.