A proof of the rooted tree alternative conjecture

TL;DR

This paper proves the rooted tree analogue of Bonato and Tardif's conjecture, showing that the number of isomorphism classes of mutually embeddable rooted trees is either one or infinite, and discusses related conjectures.

Contribution

The paper establishes the rooted tree version of the conjecture and explores related open problems for locally finite trees.

Findings

Proved the rooted tree analogue of the conjecture.

Discussed the original conjecture for locally finite trees.

Proposed new related conjectures.

Abstract

Bonato and Tardif conjectured that the number of isomorphism classes of trees mutually embeddable with a given tree T is either 1 or infinite. We prove the analogue of their conjecture for rooted trees. We also discuss the original conjecture for locally finite trees and state some new conjectures.