Uncountably many homogeneous real trees with the same valence

Pénélope Azuelos

Uncountably many homogeneous real trees with the same valence

Metric Geometry 2025-11-06 v1

Authors: Pénélope Azuelos

Abstract

For any cardinal $\kappa \geq 2$ , there is a unique complete real tree whose points all have valence $\kappa$ . In this note, we show that, when $\kappa \geq 3$ , it is necessary to assume completeness. More precisely, we show that there exist uncountably many homogeneous incomplete real trees whose points all have valence $\kappa$ .

Keywords

set theory trees phylogenetic trees

Cite

@article{arxiv.2511.03722,
  title  = {Uncountably many homogeneous real trees with the same valence},
  author = {Pénélope Azuelos},
  journal= {arXiv preprint arXiv:2511.03722},
  year   = {2025}
}