A tree has a countable vertex set exactly when some labeling makes its generated ultrametric separable or locally finite, and local finiteness is governed by rays and star subgraphs.
Extremal properties and morphisms of finite ultrametric spaces and their representing trees
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abstract
We study extremal properties of finite ultrametric spaces $X$ and related properties of representing trees $T_X$. The notion of weak similarity for such spaces is introduced and related morphisms of labeled rooted trees are found. It is shown that the finite rooted trees are isomorphic to the rooted trees of nonsingular balls of special finite ultrametric spaces. We also found conditions under which the isomorphism of representing trees $T_X$ and $T_Y$ implies the isometricity of ultrametric spaces $X$ and $Y$.
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Labeled Trees Generating Separable and Locally Finite Ultrametrics
A tree has a countable vertex set exactly when some labeling makes its generated ultrametric separable or locally finite, and local finiteness is governed by rays and star subgraphs.