It is well known that there is a tree structure associated with any discrete non-Archimedean valued field or more generally with any ultrametric space with discrete metric. In this paper, given any set with at least two elements, we study families of partitions of the set that satisfy certain conditions and introduce a tree structure on each such family where vertices are partitions in the family. Furthermore, for any ultrametric space with discrete metric that contains at least two points, we associate with that space a family of partitions that satisfies these conditions and show that the tree structure associated with the space is isomorphic to the tree structure on the family of partitions associated with the space.

graphs, non-Archimedean valued fields, partitions, trees, ultrametric spaces.