# A tree distinguishing polynomial

## TL;DR

This paper introduces a bivariate polynomial for unlabeled rooted and unrooted trees, proving it is a complete invariant for tree isomorphism, and generalizes the concept to unrooted trees.

## Contribution

The paper defines a new polynomial invariant for unlabeled trees and proves its completeness for both rooted and unrooted cases, extending previous invariants.

## Key findings

- The polynomial uniquely identifies unlabeled rooted trees.
- The polynomial extends to unrooted trees as a complete invariant.
- The polynomial serves as a generating function for certain subtrees.

## Abstract

We define a bivariate polynomial for unlabeled rooted trees and show that the polynomial of an unlabeled rooted tree $T$ is the generating function of a class of subtrees of $T$. We prove that the polynomial is a complete isomorphism invariant for unlabeled rooted trees. Then, we generalize the polynomial to unlabeled unrooted trees and we show that the generalized polynomial is a complete isomorphism invariant for unlabeled unrooted trees.