Abstract
It is shown that a locally finite ultrametric space (X, d) is generated by a labeled tree if and only if for every open ball B ⊆ X there is a point c ∈ B such that d(x, c) = diam B whenever x ∈ B and x ≠ c. For every finite ultrametric space Y, we construct an ultrametric space Z having the smallest possible number of points such that Z is generated by a labeled tree and Y is isometric to a subspace of Z. It is proved that for a given Y such a space Z is unique up to isometry.
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 20, No. 3, pp. 350–380, July–September, 2023.
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Dovgoshey, O., Kostikov, A. Locally finite ultrametric spaces and labeled trees. J Math Sci 276, 614–637 (2023). https://doi.org/10.1007/s10958-023-06786-3
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DOI: https://doi.org/10.1007/s10958-023-06786-3