Alavi–Malde–Schwenk–Erdős unimodality conjecture for independence polynomials of trees

Let $G$ be a tree, and let

$$I_G(t)=i_0+i_1t+\cdots+i_{\alpha(G)}t^{\alpha(G)}$$

be its independence polynomial, where $\alpha(G)$ is the maximum size of an independent set in $G$ and $i_j$ counts the independent sets of $G$ of cardinality $j$. A polynomial $I_G(t)$ of degree $d$ is unimodal if there is an index $0\leq j\leq d$ such that

$$i_0\leq\cdots\leq i_{j-1}\leq i_j\geq i_{j+1}\geq\cdots\geq i_d.$$

Alavi–Malde–Schwenk–Erdős conjecture. The independence polynomial of every tree is unimodal.

The conjecture was proposed in 1987. The paper establishes log-concavity, and hence unimodality, for spiders and pineapple graphs, providing further evidence, but the general tree case remains open in the supplied context.