  We consider determinantal point processes (DPPs) constrained by spanning
trees. Given a graph $G=(V,E)$ and a positive semi-definite matrix $\mathbf{A}$
indexed by $E$, a spanning-tree DPP defines a distribution such that we draw
$S\subseteq E$ with probability proportional to $\det(\mathbf{A}_S)$ only if
$S$ induces a spanning tree. We prove $\sharp\textsf{P}$-hardness of computing
the normalizing constant for spanning-tree DPPs and provide an
approximation-preserving reduction from the mixed discriminant, for which FPRAS
is not known. We show similar results for DPPs constrained by forests.
