On the roots of the subtree polynomial

TL;DR

This paper investigates the complex roots of the subtree polynomial of trees, showing they are confined within a specific disk and identifying the unique tree with a boundary root, while also analyzing real root intervals.

Contribution

It establishes bounds on the roots of subtree polynomials and characterizes the trees with roots on the boundary, providing new insights into their root distribution.

Findings

Complex roots are within the disk |z| ≤ 1 + ∛3.

Only K_{1,3} has a root on the boundary of the disk.

The real roots' closure contains the interval [-2, -1].

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 $\left\{z\in\mathbb{C}\colon\ |z|\leq 1+\sqrt[3]{3}\right\}$, 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.