Consistency of the Baryon-Multimeson Amplitudes for Large-N_c QCD

We study the pion-baryon scattering process $\pi+B \to (n-1)\pi+B$ in a QCD theory with a large number (N_c) of colors. It is known that this scattering amplitudes decreases with N_c like N_c^{1-n/2}, and that its individual tree diagrams grow like N_c^{n/2}. The only way these two can be consistent is for n-1 powers of N_c to be cancelled when the Feynman diagrams are summed. We prove this to be true in tree order for any n.