Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a BQO. Naturally, two interesting questions arise:
1. What is the number \lambda of topological types of locally finite trees?
2. What are the possible sizes of an equivalence class of locally finite trees?
For (1), clearly, \omega_0 \leq \lambda \leq c and Matthiesen refined it to \omega_1 \leq \lambda \leq c. Thus, this question becomes non-trivial in the absence of the Continuum Hypothesis. In this paper we address both questions by showing - entirely within ZFC - that for a large collection of locally finite trees that includes those with countably many rays:
- \lambda = \omega_1, and
- the size of an equivalence class can only be either 1 or c.
- Analytic Topology in Mathematics and Computer Science