We prove factorization of the generating functional of connected tree diagrams by exploring that it is the Legendre transform of the action. This theorem is then applied to the example of a local relativistic interacting field theory in 2D with a single massive real scalar field that has no derivative couplings and no classical tadpole. In the process we streamline the proof that the assumption of no particle production leads to either the sin(h)-Gordon or the Bullough-Dodd model.