We give a framework for growth models on posets which simultaneously generalizes the Classical Sequential Growth models for posets from causal set theory and the tree growth models of natural growth and simple tree classes, the latter of which also appear as solutions of combinatorial Dyson-Schwinger equations in quantum field theory. We prove which cases of the Classical Sequential Growth models give subHopf algebras of the Hopf algebra of posets, in analogy to a characterization due to Foissy in the Dyson-Schwinger case. We find a family of generating sets for the Connes-Moscovici Hopf algebra.