Uncountably many homogeneous real trees with the same valence

TL;DR

This paper demonstrates that for any valence , there are uncountably many homogeneous incomplete real trees with that valence, contrasting with the unique complete case, highlighting the richness of incomplete structures.

Contribution

It shows the existence of uncountably many homogeneous incomplete real trees with a given valence , extending the classification beyond the complete case.

Findings

Uncountably many homogeneous incomplete real trees exist for  .

Completeness is necessary for uniqueness when  .

Complete real trees are unique for each valence .

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 $\kappa$.