  Some years ago Caporaso and Harris have found a nice way to compute the
numbers N(d,g) of complex plane curves of degree d and genus g through 3d+g-1
general points with the help of relative Gromov-Witten invariants. Recently,
Mikhalkin has found a way to reinterpret the numbers N(d,g) in terms of
tropical geometry and to compute them by counting certain lattice paths in
integral polytopes. We relate these two results by defining an analogue of the
relative Gromov-Witten invariants and rederiving the Caporaso-Harris formula in
terms of both tropical geometry and lattice paths.

Algebraic Geometry