Motivated by the classification problem of left invariant locally flat affine structures on Lie groups, Segal proved in 1994 a version of the Poincaré-Birkhoff-Witt for universal enveloping pre-Lie algebras of Lie algebras; these algebras were studied in more detail by Bolgar in 1996. Recent work of the first author and Tamaroff implies that a stronger version of such theorem holds, meaning that the PBW isomorphisms have a strong functoriality property. By contrast with the classical PBW theorem, neither of the abovementioned results leads to a description of the Schur functor one has to use to compute the underlying space of the universal enveloping algebra. In this paper, we compute the corresponding Schur functor in terms of combinatorics of rooted trees.