  In this paper, we give a tropical method for computing Gromov-Witten type
invariants of Fano manifolds of special type. This method applies to those Fano
manifolds which admit toric degenerations to toric Fano varieties with
singularities allowing small resolutions. Examples include (generalized) flag
manifolds of type A, and some moduli space of rank two bundles on a genus two
curve.
