Zig-zag chains and metric equivalences between ultrametric spaces

We study the classification of ultrametric spaces based on their small scale geometry (uniform homeomorphism), large scale geometry (coarse equivalence) and both (bi-uniform equivalences). Using a combinatoric approach we consider every ultrametric space as the end space of a chain and prove that all these equivalences can be characterized by the existence of a common zig-zag chain.