  We present a set of integer programs (IPs) for the Steiner tree problem with
the property that the best solution obtained by solving all, provides an
optimal Steiner tree. Each IP is polynomial in the size of the underlying graph
and our main result is that the linear programming (LP) relaxation of each IP
is integral so that it can be solved as a linear program. However, the number
of IPs grows exponentially with the number of terminals in the Steiner tree. As
a consequence, we are able to solve the Steiner tree problem by solving a
polynomial number of LPs, when the number of terminals is fixed.
