Totally bounded ultrametric spaces and locally finite trees

We investigate the interrelations between the metric properties, order properties and combinatorial properties of the set of balls in totally bounded ultrametric space. In particular, the Gurvich-Vyalyi representation of finite, ultrametric spaces by monotone rooted trees is generalized to the case of totally bounded ultrametric spaces. It is shown that such spaces have isometric completions if and only if their labeled representing trees are isomorphic. We characterize up to isomorphism the representing trees of these spaces and, up to order isomorphism, the posets of open balls in such spaces.