  In this paper we introduce broccoli curves, certain plane tropical curves of
genus zero related to real algebraic curves. The numbers of these broccoli
curves through given points are independent of the chosen points - for
arbitrary choices of the directions of the ends of the curves, possibly with
higher weights, and also if some of the ends are fixed. In the toric Del Pezzo
case we show that these broccoli invariants are equal to the Welschinger
invariants (with real and complex conjugate point conditions), thus providing a
proof of the independence of Welschinger invariants of the point conditions
within tropical geometry. The general case gives rise to a tropical
Caporaso-Harris formula for broccoli curves which suffices to compute all
Welschinger invariants of the plane.
