Two graphs are of the same topological type if they can be mutually embedded into each other topologically. We show that there are exactly ₁ ℵ 1 distinct topological types of countable trees. In general, for any infinite cardinal κ κ there are exactly κ ⁺ κ + distinct topological types of trees of size κ κ . This solves a problem of van der Holst from 2005.
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Krill et al. (2024) studied this question.