if a group g acts on a tree x we may determine the precise structure of g byapplying machinery of bass and serre for reconstructing the group action. we use this to give structure theorems for certain subgroups of $sl_2 $ over a on-archimedean local field, and of groups associated to kac-moody lie algebras over finite fields. of special interest are lattice subgroups and their congruence subgroups.