Starlike rayless tree characterization via completions of labeled rays

Establish the equivalence for every infinite ultrametric space (X, d) that: (i) (X, d) is the completion of a totally bounded subspace X0 ⊂ X generated by a labeled ray; and (ii) there exists a starlike rayless tree T with a labeling l: V(T) → R+ such that (X, d) is a compact ultrametric space generated by the labeled tree T(l).

Background

A starlike tree is a tree with exactly one vertex of degree greater than 2; rayless means it has no infinite ray. The paper proves (Theorem 5.6) that compact US-spaces (those generated by labeled star graphs) are precisely the completions of totally bounded ultrametric spaces generated by decreasingly labeled rays. Conjecture 6.2 proposes a broader equivalence replacing star graphs by starlike rayless trees, thereby extending Theorem 5.6 beyond the US-class.

This conjecture seeks a structural correspondence between compact ultrametric spaces generated by labeled starlike rayless trees and completions of ultrametric spaces generated by labeled rays. It would generalize the star-graph case to a larger family of generating trees.