  We show that any 2-valued C^{1, \alpha} (\alpha \in (0, 1)) function u =
{u_{1}, u_{2}} on an open ball B in {\mathbb R}^{n} with values u_{1}, u_{2}
\in {\mathbb R}^{k} whose graph, viewed as a varifold with multiplicity 2 at
points where u_{1} = u_{2} and with multiplicity 1 at points where u_{1}, u_{2}
are distinct, is stationary in the cylinder B \times {\mathbb R}^{k} must be a
C^{1, 1/2} function, and the set of its branch points, if non-empty, must have
Hausdorff dimension (n-2) and locally positive (n-2)-dimensional Hausdorff
measure. The C^{1, 1/2} regularity is optimal.
