  Enumerative algebraic geometry deals with problems of counting geometric
objects defined algebraically, An important class of enumerative problems is
that of counting curves: given a class of curves in some projective variety
defined by fixing some algebraic or geometric invariants (such as degree, genus
and types of singularities), the problem usually takes the form of "how many
curves of that class pass through a configuration of n points in general
position?" Tropical Geometry deals with certain piecewise-linear complexes,
which arise as degeneration of families of complex algebraic varieties, and can
also be described algebraically using "max-plus" algebra, (The tropical
semi-field). The problem we solve is that of counting rational curves with one
cusp and certain number of nodes on toric surfaces, passing through a
configuration of sufficient points in general position. We show that that this
number equals the number of certain tropical curves counted with multiplicities
and we describe these curves and their multiplicities. The main tools are
tropicalization and patchworking. In tropicalization we pass from an
equisingular family of curves to a special limit fiber which can be described
in terms of tropical data and analytic data. We then classify these possible
limits, and use the patchworking theorem to reconstruct the families that
correspond to them.
