We exhibit relations between enumerative invariants of the
Hirzebruch surfaces $\Sigma_n$ and
\Sigma_{n+2}, obtained by
deforming the first surface to the latter.
Our formula generalizes in particular
a formula due to Abramovich and Bertram, extended later by Vakil.
We use methods from tropical
geometry,
involving
a tropical counterpart of
deformations of Hirzebruch
surfaces, and tropical enumerative geometry on a tropical
surface in three-space.