{
  "abstract": "For a tree $T$, let $[T]$ be the set of isomorphism classes of trees that are mutually topological minors of $T$. Bruno and Szeptycki proved that every locally finite tree satisfies $|[T]|\\in\\{1,2^{\\aleph_0}\\}$ and that every tree with a ray containing infinitely many vertices of degree at least $3$ has at least $2^{\\aleph_0}$ topological twins. Hence equality holds in the latter case when the tree is countable. We treat the complementary countable case, without any bound on the degrees. For every such tree $T$, we prove that $|[T]|\\in\\{1,\\aleph_0,2^{\\aleph_0}\\}$ and give a well-founded recursive description based on Schmidt rank. The proof uses a finite canonical subtree extracted from the subtree generated by the branching vertices and a counting theorem for countable multisets over a well-quasi-order. The same trichotomy follows for every countable tree and hence for every tree all of whose vertices have countable degree.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27872",
  "author_total": 2,
  "authors": [
    {
      "name": "Qi Wu"
    },
    {
      "name": "Yong Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/86a0ap3b1xgww.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27872",
  "fulltext_url": null,
  "id": "86a0ap3b1xgww",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27872",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7214166102",
  "pdf_url": "https://arxiv.org/pdf/2609.27872",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/86a0ap3b1xgww.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27872",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27872"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "A cardinal trichotomy for topological equivalence classes of countable trees",
  "title_text": "A cardinal trichotomy for topological equivalence classes of countable trees",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/86a0ap3b1xgww",
  "venue": "arXiv",
  "versions": []
}