A Proof of the Tree Alternative Conjecture Under the Topological Minor Relation

We prove the Tree Alternative Conjecture for the topological minor relation: letting $[T]$ denote the equivalence class of $T$ under the topological minor relation we show that: $|[T]| = 1$ or $|[T]|\geq \aleph_0$ and $\forall r\in V(T)$, $|[(T,r)]| = 1$ or $|[(T,r)]|\geq \aleph_0$. In particular, by means of curtailing trees, we show that for any tree $T$ with at least one not eventually bare ray: $|[T]| \geq 2^{\aleph_0}$.