Ordered Forests, Permutations, and Iterated Integrals

We construct an explicit Hopf algebra isomorphism from the algebra of heap-ordered trees to that of quasi-symmetric functions, generated by formal permutations, which is a lift of the natural projection of the Connes–Kreimer algebra of decorated rooted trees onto the shuffle algebra. This isomorphism gives a universal way of lifting measure-indexed characters of the Connes–Kreimer algebra into measure-indexed characters of the shuffle algebra, already introduced in Unterberger [Hölder-continuous rough paths by Fourier normal ordering, Communications in Mathematical Physics 298, no. 1 (2010) 1–36] in the framework of rough path theory as the so-called Fourier normal ordering algorithm.