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.
Springer, Berlin (2016)
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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.