  In 1983 Kalai proved an incredible generalisation of Cayley's formula for the
number of trees on a labelled vertex set to a formula for a class of
$r$-dimensional simplicial complexes. These simplicial complexes generalise
trees by means of being homologically $\Bbb Q$-acyclic. In this text we
consider a different generalisation of trees to the class of pure dimensional
simplicial complexes that \textit{minimally connect a vertex set} (in the sense
that the removal of any top dimensional face disconnects the complex). Our main
result provides an upper and lower bound for the number of these minimally
connected complexes on a labelled vertex set. We also prove that they are
potentially vastly more topologically complex than the generalisation of Kalai.
As an application of our bounds we compute the threshold probability for the
connectivity of a random simplicial complex.
