A proof of the rooted tree alternative conjecture

Bonato and Tardif [A. Bonato, C. Tardif, Mutually embeddable graphs and the tree alternative conjecture, J. Combinatorial Theory, Series B 96 (2006), 874–880] conjectured that the number of isomorphism classes of trees mutually embeddable with a given tree TT is either 1 or infinite. We prove the analogue of their conjecture for rooted trees. We also make some progress towards the original conjecture for locally finite trees and state some new conjectures.