  We exhibit an explicit natural isomorphism between spaces of branched and
geometric rough paths. This provides a multi-level generalisation of the
isomorphism of Lejay-Victoir (2006) as well as a canonical version of the
It\^o-Stratonovich correction formula of Hairer-Kelly (2015). Our construction
is elementary and uses the property that the Grossman-Larson algebra is
isomorphic to a tensor algebra. We apply this isomorphism to study signatures
of branched rough paths. Namely, we show that the signature of a branched rough
path is trivial if and only if the path is tree-like, and construct a
non-commutative Fourier transform for probability measures on signatures of
branched rough paths. We use the latter to provide sufficient conditions for a
random signature to be determined by its expected value, thus giving an answer
to the uniqueness moment problem for branched rough paths.
