Gao and Kechris proposed in 2003 two somewhat related problems concerning ultrametric spaces, namely:
1) Determine the complexity of the isometry relation on locally compact Polish ultrametric spaces.
2) Characterize the Polish groups that are isomorphic (as topological groups) to the isometry group of some Polish ultrametric space.
We will present a construction strictly relating ultrametric spaces and a special kind of trees which helps in tackling these two problems. This technique applies to both separable and non-separable complete ultrametric spaces, and allows us to e.g. show that they are unclassifyiable up to isometry even when considering only discrete spaces. (Joint work with R. Camerlo and A. Marcone.)