  We study three combinatorial models for the lower-triangular matrix with
entries $t_{n,k} = \binom{n}{k} n^{n-k}$: two involving rooted trees on the
vertex set $[n+1]$, and one involving partial functional digraphs on the vertex
set $[n]$. We show that this matrix is totally positive and that the sequence
of its row-generating polynomials is coefficientwise Hankel-totally positive.
We then generalize to polynomials $t_{n,k}(y,z)$ that count improper and proper
edges, and further to polynomials $t_{n,k}(y,\mathbf{\phi})$ in infinitely many
indeterminates that give a weight $y$ to each improper edge and a weight $m! \,
\phi_m$ for each vertex with $m$ proper children. We show that if the weight
sequence $\mathbf{\phi}$ is Toeplitz-totally positive, then the two foregoing
total-positivity results continue to hold. Our proofs use production matrices
and exponential Riordan arrays.
