Considering the iterated integral \[ I^{ts}_\Gamma(a_1,\ldots,a_n)=\int^t_s\text d\Gamma_{a_1}(x_1)\int^t_s\text d\Gamma_{a_2}(x_2)\cdots\int^t_s\text d\Gamma_{a_n}(x_n), \] where \(\Gamma_1,\ldots,\Gamma_d\) are \(d\) regular functions, \(s,t\in\mathbb R\) and \(a_1,\ldots,a_n\) is a word in the letters \(\{1,\ldots,d\}\), one can extend it from words to decorated rooted trees. And inversely each such extension can be decomposed into a sum of integrals of specific words.
Algebraically this process corresponds to a Hopf algebra morphism from the Hopf algebra of decorated rooted trees to the Hopf algebra generated by words. This map can be lifted to a Hopf algebra morphism \(\Theta\) from heap ordered decorated forests \(\mathbf H^d_{ho}\) to quasi-symmetric functions \(\mathbf{FQSym}^d\). The authors also compute the preimage by \(\Theta\) of any element of \(\mathbf{FQSym}^d\) and interpret their algebraic construction in rough path theory.