On the roots of the subtree polynomial

Abstract

For a tree $T$, the subtree polynomial of $T$ is the generating polynomial for the number of subtrees of $T$. We show that the complex roots of the subtree polynomial are contained in the disk $\left\{z\in \mathbb{C}:|z|\le 1+\sqrt[3]{3}\right\}$, and that $K_{1,3}$ is the only tree whose subtree polynomial has a root on the boundary. We also prove that the closure of the collection of all real roots of subtree polynomials contains the interval $[-2,-1]$, while the intervals $(\infty ,-1-\sqrt[3]{3})$, $[-1,0)$, and $(0,\infty )$ are root-free.

16 pages, 3 figures, comments welcome