  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 \cite{Unterberger} in the framework of rough path theory
as the so-called Fourier normal ordering algorithm.
