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 $t_{n,k}(y,\mathbfφ)$ in infinitely many indeterminates that give a weight $y$ to each improper edge and a weight $m! \, φ_m$ for each vertex with $m$ proper children. We show that if the weight sequence $\mathbfφ$ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.