What is the distance in the sense of Gromov-Hausdorff between $\mathbb{Z}\_{p\_1}$ and $\mathbb{Z}\_{p\_2}$ with the usual p-adic metrics?
I got stuck and simply have no idea how to deal with such questions: I've got two metric trees and have to observe somehow all embeddings to all spaces which seems a bit intractable.



Gromov-Hausdorff distance between p-adic integers.
The Gromov--Hausdorff distance is good only to define topology;
i.e., one should not care about distance between particular spaces.
But since you insist, I will answer an easier question which is closely related. 


There is a modified distance $d'\_{GH}(X,Y)$ defined as infimum of all numbers $\varepsilon>0$ such that there are maps $f\_1\colon X\to Y$ and $f\_2\colon Y\to X$ such that
$$|f\_i(x)-f\_i(y)|\ge |x-y|-\varepsilon.$$ 


This distance $d^\prime\_{GH}$ is equivalent to $d\_{GH}$
and it is usually easier to find value $d^\prime\_{GH}$


If $ p < q < p^2$ then it is easy to see that


$$ d^\prime\_{GH} ( \mathbb Z\_{p},\mathbb Z\_{q}) = \tfrac{p-1}{p}. $$
Further, if $ p^2 < q < p^3$ then


$$ d^\prime\_{GH} ( \mathbb Z\_{p},\mathbb Z\_{q}) = \tfrac{p^2-1}{p^2}$$
and so on.


