  We use the diagram-free approach to regularity structures introduced by Otto
et. al. to build rough paths based on multi-indices. We identify the analogue
of the insertion pre-Lie algebra of trees and use it to build the corresponding
group of translations of rough paths. We make this identification precise by
showing that the natural dictionary between trees and multi-indices is a
pre-Lie morphism under insertion, which in turn yields a Hopf algebra morphism
to the rooted tree Hopf algebra equipped with the extraction-contraction
coproduct.
