Two graphs are of the same topological type if they can be mutually embedded into each other topologically. We show that there are exactly \(\aleph _1\) distinct topological types of countable trees. In general, for any infinite cardinal \(\kappa \) there are exactly \(\kappa ^+\) distinct topological types of trees of size \(\kappa \). This solves a problem of van der Holst from 2005.

中文翻译：

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树的拓扑类型数量

如果两个图可以在拓扑上相互嵌入，则它们具有相同的拓扑类型。我们证明，可数树确实存在\(\aleph _1\)个不同的拓扑类型。一般来说，对于任何无限基数\(\kappa \) ，都存在\(\kappa ^+\)大小为\(\kappa \)的不同拓扑类型的树。这解决了 2005 年 van der Holst 的问题。