In tree level Yang-Mills theory and Einstein gravity on-shell recursion relations have been proven which allow one to express any amplitude in terms of certain sums over three point amplitudes. These relations look remarkably like CFT bootstrap equations and are closely related to crossing symmetry. In this talk it will be shown how these recursion relations can be proven for string amplitudes on the disc and sphere in a flat background. A completely CFT based argument will be presented. This suggests that the techniques explored may extend to a much more general setting in string theory. Some simple examples of this will be discussed.