  We prove two tropical gluing formulae for Gromov-Witten invariants of
exploded manifolds, useful for calculating Gromov-Witten invariants of a
symplectic manifold using a normal-crossing degeneration. The first formula
generalizes the symplectic-sum formula for Gromov-Witten invariants. The second
formula is stronger, and also generalizes Kontsevich and Manin's splitting and
genus-reduction axioms. Both tropical gluing formulae have versions
incorporating gravitational descendants.
