  We deduce a structurally inductive description of the determinantal
probability measure associated with Kalai's celebrated enumeration result for
higher--dimensional spanning trees of the $n-1$--simplex. As a consequence, we
derive the marginal distributions of the simplex links in such random trees.
Along the way, we also characterize the higher--dimensional spanning trees of
every other simplicial cone in terms of the higher--dimensional rooted forests
of the underlying simplicial complex. We also apply these new results to random
topology, the spectral analysis of random graphs, and the theory of high
dimensional expanders. One particularly interesting corollary of these results
is that the fundamental group of a union of $o(\log n)$ determinantal 2--trees
has Kazhdan's property (T) with high probability.
