  Let $N$ be a Riemannian manifold and consider a stationary union of three or
more $C^{1,\mu}$ hypersurfaces-with-boundary $M_k$ in $N$ with a common
boundary $\Gamma$. We show that if $N$ is smooth, then $\Gamma$ is smooth and
each $M_k$ is smooth up to $\Gamma$ (real analytic in the case $N$ is real
analytic). Consequently we strengthen a result of Wickramasekera to conclude
that under the stronger hypothesis that $V$ is a stationary, stable, integral
$n$-varifold in an $(n+1)$-dimensional, smooth (real analytic) Riemannian
manifold such that the support of $\|V\|$ is nowhere locally the union of three
or more smooth (real analytic) hypersurfaces-with-boundary meeting along a
common boundary, the singular set of $V$ is empty if $n = 6$, is discrete if $n
= 7$, and has Hausdorff dimension at most $n-7$ if $n \geq 8$.
