  The aim of this paper is to bring together the three objects in the title.
Recall that, given a Lie algebra $\mathfrak{g}$, the Eulerian idempotent is a
canonical projection from the enveloping algebra $U(\mathfrak{g})$ to
$\mathfrak{g}$. The Baker-Campbell-Hausdorff product and the Magnus expansion
can both be expressed in terms of the Eulerian idempotent, which makes it
interesting to establish explicit formulas for the latter. We show how to
reduce the computation of the Eulerian idempotent to the computation of a
logarithm in a certain pre-Lie algebra of planar, binary, rooted trees. The
problem of finding formulas for the pre-Lie logarithm, which is interesting in
its own right -- being related to operad theory, numerical analysis and
renormalization -- is addressed using techniques inspired by umbral calculus.
As a consequence of our analysis, we find formulas both for the Eulerian
idempotent and the pre-Lie logarithm in terms of the combinatorics of trees.
