  The pseudodeterminant $\textrm{pdet}(M)$ of a square matrix is the last
nonzero coefficient in its characteristic polynomial; for a nonsingular matrix,
this is just the determinant. If $\partial$ is a symmetric or skew-symmetric
matrix then $\textrm{pdet}(\partial\partial^t)=\textrm{pdet}(\partial)^2$.
Whenever $\partial$ is the $k^{th}$ boundary map of a self-dual CW-complex $X$,
this linear-algebraic identity implies that the torsion-weighted generating
function for cellular $k$-trees in $X$ is a perfect square. In the case that
$X$ is an \emph{antipodally} self-dual CW-sphere of odd dimension, the
pseudodeterminant of its $k$th cellular boundary map can be interpreted
directly as a torsion-weighted generating function both for $k$-trees and for
$(k-1)$-trees, complementing the analogous result for even-dimensional spheres
given by the second author. The argument relies on the topological fact that
any self-dual even-dimensional CW-ball can be oriented so that its middle
boundary map is skew-symmetric.
