The problem is to embed Cayley graph of free group with $n\geq2$ generators (the same as Bethe lattice with coordination number $2n$) into any model of $\mathbb{H}^2$ (we have no model preference, the only condition is to preserve the metric structure of the graph). Any numerical algorithms like MDS are not suitable. Unfortunately, I can't find any explicit formulas. But my guess is that insofar as embedding is unique, formula must exist. I would be glad if someone help with useful papers or provide any useful reasoning.

UPD Two additions: 1) embedding must be isomorphic, 2) I meant the embedding into $\mathbb{H}^n$ – I guess that for $n>2$ generators Cayley graph cannot be embedded into hyperbolic plane.