A family of ω₁ topological types of locally finite trees

Two rooted locally finite trees are considered equivalent if both can be embedded into each other as topological minors by means of tree-order preserving mappings. By exploiting Nash-William's Theorem, Matthiesen provided a non-constructive proof of the uncountability of such equivalence classes, thus answering a question of van der Holst. As an open problem, Matthiesen asks for a constructive proof of this fact. The purpose of this paper is to provide one such construction; working solely within ZFC we illustrate a collection of ω₁ many topological types of rooted trees. In particular, we also show that this construction strengthens that of Matthiesen in that it also applies to free (unrooted) trees of degree two.