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.