I will discuss the "on-shell constructibility" of tree amplitudes from recursion relations in general 4-dimensional local field theories with any type of particles. This analysis applies to renormalizable as well as non-renormalizable interactions, with or without supersymmetry. The focus will be on recursion relations that arise from complex deformations of all external momenta. Under certain conditions, these "all-line shift recursion relations" imply the MHV vertex expansion. I will derive a simple sufficient criterion for the validity of the all-line shift recursion relations. It depends only on the mass dimensions of the coupling constants and on the sum of helicities of the external particles. This proof is strikingly simple since it just relies on dimensional analysis and little-group transformation properties. In particular, I will demonstrate that all tree amplitudes with n>4 external states are constructible in any power-counting renormalizable theory. Aspects of all-line shift constructibility will be illustrated in numerous examples, ranging from pure scalar theory to theories with higher-derivative interactions. I will conclude with a sharp physical interpretation of the constructibility criterion: the all-line shift fails precisely for those classes of n-point amplitudes that can receive local contributions from independent gauge-invariant n-field operators.