  Iterated integrals of paths arise frequently in the study of the Taylor's
expansion for controlled differential equations. We will prove a factorial
decay estimate, conjectured by M. Gubinelli, for the iterated integrals of
non-geometric rough paths. We will explain, with a counter example, why the
conventional approach of using the neoclassical inequality fails. Our proof
involves a concavity estimate for sums over rooted trees and a non-trivial
extension of T. Lyons' proof in 1994 for the factorial decay of iterated
Young's integrals.
