  We extend the recently developed rough path theory for Volterra equations
from (Harang and Tindel, 2019) to the case of more rough noise and/or more
singular Volterra kernels. It was already observed in (Harang and Tindel, 2019)
that the Volterra rough path introduced there did not satisfy any geometric
relation, similar to that observed in classical rough path theory. Thus, an
extension of the theory to more irregular driving signals requires a deeper
understanding of the specific algebraic structure arising in the Volterra rough
path. Inspired by the elements of "non-geometric rough paths" developed in
(Gubinelli, 2010) and (Hairer and Kelly, 2015) we provide a simple description
of the Volterra rough path and the controlled Volterra process in terms of
rooted trees, and with this description we are able to solve rough volterra
equations in driven by more irregular signals.
