  We obtain a complete time expansion of the pull-back operator generated by a
real analytic flow of real analytic automorphisms acting on analytic tensor
sections of a manifold. Our expansion is given in terms of multiple Lie
derivatives. Motivated by this expansion, we provide a rather simple and
explicit estimate for higher order covariant derivatives of multiple Lie
derivatives acting on smooth endomorphism sections of the tangent bundle of a
manifold. We assume the covariant derivative to be torsion free. The estimate
is given in terms of Dyck polynomials. The proof uses a new result on the
combinatorics of rooted labeled ordered forests and Dyck polynomials.
