We develop the framework of exotic forests to study the algebraic structures underlying numerical integrators for sampling the invariant measure of Langevin dynamics. Leveraging these algebraic structures, we introduce a formal algorithm for generating order conditions for invariant measure sampling and prove that a large subset of these conditions is satisfied automatically. Building on these insights, we derive an efficient second-order integrator for sampling the invariant measure of Langevin dynamics with position-dependent diffusion.

We then consider a broader set of exotic forests that includes graph structures known as aromas and stolons, which play a crucial role in geometric numerical integration. We analyze their associated composition and substitution laws, enabling the study of integrator composition, post-processing, backward error analysis, and modified equation techniques within the Butcher series framework. Given the rapidly increasing complexity of the forest structures we consider, we develop a Haskell package to automate computations involving algebras over forests.