  For a relative effective divisor $\mathcal{C}$ on a smooth projective family
of surfaces $q:\mathcal{S}\rightarrow B$, we consider the locus in $B$ over
which the fibres of $\mathcal{C}$ are $\delta$-nodal curves. We prove a
conjecture by Kleiman and Piene on the univerality of an enumerating cycle on
this locus. We propose a bivariant class $\gamma(\mathcal{C})\in A^*(B)$
motivated by the BPS calculus of Pandharipande and Thomas, and show that it can
be expressed universally as a polynomial in classes of the form
$q_*(c_1(\mathcal{O}(\mathcal{C}))^a c_1(T_{\mathcal{S}/B})^b
c_2(T_{\mathcal{S}/B})^c)$. Under an ampleness assumption, we show that
$\gamma(\mathcal{C})\cap[B]$ is the class of a natural effective cycle with
support equal to the closure of the locus of $\delta$-nodal curves. Finally, we
will apply our method to calculate node polynomials for plane curves
intersecting general lines in $\mathbb{P}^3$. We verify our results using 19th
century geometry of Schubert.
