The Tree Alternative Conjecture Under the Topological Minor Relation

Abstract

The Tree Alternative Conjecture concerns the sizes of equivalence classes of trees with respect to mutual embeddable relation. Indeed, it conjectures that the number of isomorphism classes of trees mutually embeddable with a given tree $T$ is either 1 or infinite - with instances of size $\aleph_0$ and $2^{\aleph_0}$. We prove its analogue within the family of locally finite trees with respect to the topological minor relation. More precisely, we prove that for any locally finite tree $T$ the size of its equivalence class with respect to the topological minor relation can only be either $1$ or $2^{\aleph_0}$.