  Elements of the tropical vertex group, introduced by Kontsevich and
Soibelman, are formal families of symplectomorphisms of the 2-dimensional
algebraic torus. We prove ordered product factorizations in the tropical vertex
group are equivalent to calculations of certain genus 0 relative Gromov-Witten
invariants of toric surfaces. The relative invariants which arise have full
tangency to a toric divisor at a single unspecified point. The method uses
scattering diagrams, tropical curve counts, degeneration formulas, and exact
multiple cover calculations in orbifold Gromov-Witten theory.
