  In the 1980's, Almgren developed a theory of multi-valued Dirichlet energy
minimizing functions on $n$ dimensional domains and used it, in an essential
way, to bound the Hausdorff dimension of the singular sets of area minimizing
rectifiable currents of dimension $n$ and codimension $\geq 2$. Recent work of
the second author shows that two-valued $C^{1, \mu}$ harmonic functions on $n$
dimensional domains, which are typically-non-minimizing stationary points of
Dirichlet energy, play an essential role in the study of multiplicity 2 branch
points of stable codimension 1 rectifiable currents of dimension $n$. In all of
these cases (of multi-valued harmonic functions and minimal currents), it is
known that the branch sets have Hausdorff dimension $\leq n-2.$ In this paper
we initiate a study of the local structure of branch sets. We show that the
branch set of a two-valued Dirichlet energy minimizing function or a two-valued
$C^{1, \mu}$ harmonic function, in each closed ball of its domain, is either
empty or has positive $(n-2)$-dimensional Hausdorff measure and is equal to the
union of a finite number of locally compact, locally $(n-2)$-rectifiable sets.
Our method is inspired by the work of L. Simon on the structure of
singularities of minimal submanifolds in compact, multiplicity 1 classes.
