  We give an algorithm to explicitly compute the largest subtree, in the local
Bruhat-Tits tree for PSL_2(k), whose vertices correspond to orders containing a
given suborder H, in terms of a set of generators for H. The shape of this
subtree is described, when it is finite, by a set of two invariants. We use our
method to provide a full table for the invariants of an order generated by a
pair of orthogonal pure quaternions. In a previous work, the first author
showed that determining the shape of these local subtrees allows the
computation of representation fields, a class field determining the set of
isomorphism classes, in a genus O of orders of maximal rank in a fixed central
simple algebra, containing an isomorphic copy of H. Some further applications
are described here.
