  In this paper, we show how one can view certain models in regularity
structures as some form of geometric rough paths. This is performed by
identifying the deformed Butcher-Connes-Kreimer Hopf algebra with a quotient of
the shuffle Hopf algebra which is the structure underlying the definition of a
geometric rough path. This provides an extension of the isomorphism between the
Butcher-Connes-Kreimer Hopf algebra and the shuffle Hopf algebra. This new
algebraic result relies strongly on the deformation formalism and the post-Lie
structures introduced recently in the context of regularity structures.
