  We illustrate the use of tropical methods by generalizing a formula due to
Abramovich and Bertram, extended later by Vakil. Namely, 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 strategy involves a tropical counterpart of deformations of Hirzebruch
surfaces, and tropical enumerative geometry on a tropical surface in
three-space.
