Proof of the MHV vertex expansion for all tree amplitudes in N = 4 SYM theory

We prove the MHV vertex expansion for all tree amplitudes of N = 4 SYM theory. The proof uses a shift acting on all external momenta, and we show that every NkMHV tree amplitude falls off as 1/zk, or faster, for large z under this shift. The MHV vertex expansion allows us to derive compact and efficient generating functions for all NkMHV tree amplitudes of the theory. We also derive an improved form of the anti-NMHV generating function. The proof leads to a curious set of sum rules for the diagrams of the MHV vertex expansion.