Unimodality of independence polynomials of trees

Prove that the independence polynomial of every tree is unimodal, thereby resolving the Alavi–Malde–Schwenk–Erdős conjecture.

Background

For a graph, the independence polynomial enumerates independent sets by cardinality. A polynomial is unimodal when its coefficients first weakly increase and then weakly decrease. The paper recalls a conjecture posed by Alavi, Malde, Schwenk, and Erdős in 1987 asserting that every tree has a unimodal independence polynomial.

The paper establishes the stronger property of log-concavity for spiders, a particular family of trees, and presents this as further evidence for the conjecture. It does not resolve the conjecture for arbitrary trees.