We prove that the set of n-point configurations for which solution of the
planar Steiner problem is not unique has Hausdorff dimension is at most 2nβ1.
Moreover, we show that the Hausdorff dimension of n-points configurations on
which some locally minimal trees have the same length is also at most 2nβ1.
Methods we use essentially requires some analytic structure and some
finiteness, so that we prove a similar result for a complete Riemannian
analytic manifolds under some apriori assumption on the Steiner problem on
them