Palindromicity and unimodality of arbor-polytope lattice-point polynomials

Prove that, for every arbor \(\tau\), the polynomial \(h(\tau,t)\) counting lattice points in the arbor polytope \(\mathcal{Q}_\tau\) by the number of nonzero coordinates is palindromic and unimodal.

Background

For an arbor $\tau$ of size $n$, the polynomial $h(\tau,t)$ is defined by summing $t^{|\{i\in[n]:x_i\neq 0\}|}$ over all integer lattice points $x$ in the arbor polytope $\mathcal{Q}_\tau$. The paper notes that Chapoton introduced these polynomials and conjectured their palindromicity and unimodality for arbitrary arbors.

The paper establishes the stronger property of $\gamma$-positivity for octopuses and lopsided octopuses, which consequently implies palindromicity and unimodality for those classes. It does not resolve the conjecture for all arbors.