  We prove that the set of $n$-point configurations for which the solution of
the planar Steiner problem is not unique has the Hausdorff dimension at most
$2n-1$ (as a subset of $\mathbb{R}^{2n}$). Moreover, we show that the Hausdorff
dimension of the set of $n$-point configurations on which at least two locally
minimal trees have the same length is also at most $2n-1$. Methods we use
essentially require rely upon the theory of subanalytic sets developed
in~\cite{bierstone1988semianalytic}. Motivated by this approach we develop a
general setup for the similar problem of uniqueness of the Steiner tree where
the Euclidean plane is replace by an arbitrary analytic Riemannian manifold
$M$. In this setup we argue that the set of configurations possessing two
locally-minimal trees of the same length either has the dimension $n\dim M-1$
or has a non-empty interior. We provide an example of a two-dimensional surface
for which the last alternative holds.
  In addition to abovementioned results, we study the set of set of $n$-point
configurations for which there is a unique solution of the Steiner problem in
$\mathbb{R}^d$. We show that this set is path-connected.
