  We consider matrices with entries that are polynomials in $q$ arising from
natural $q$-generalisations of two well-known formulas that count: forests on
$n$ vertices with $k$ components; and trees on $n+1$ vertices where $k$
children of the root are smaller than the root. We give a combinatorial
interpretation of the corresponding statistic on forests and trees and show,
via the construction of various planar networks and the
Lindstr\"om-Gessel-Viennot lemma, that these matrices are coefficientwise
totally positive. We also exhibit generalisations of the entries of these
matrices to polynomials in \emph{eight} indeterminates, and present some
conjectures concerning the coefficientwise Hankel-total positivity of their
row-generating polynomials.
