We consider a family of quasilinear SPDEs and parameterize it in terms of the nonlinearity. We use this parameterization to provide an inductive construction of a model, which reveals the algebraic structure of the corresponding model space. We then consider the infinitesimal generators of actions in the space of nonlinearities and use them to build a (pre-)Lie algebra which gives rise to the structure group via its universal envelope. Although the approach is tree-free, we show morphism properties with respect to well-known tree-based structures in branched rough paths and regularity structures. Based on joint work with Felix Otto and Markus Tempelmayr.