  We consider the topology of real no $k$-equal spaces via the theory of
cellular spanning trees. Our main theorem proves that the rank of the
$(k-2)$-dimensional homology of the no $k$-equal subspace of $\mathbb{R}$ is
equal to the number of facets in a $k$-dimensional spanning tree of the
$k$-skeleton of the $n$-dimensional hypercube.
