  W. Magnus introduced a particular differential equation characterizing the
logarithm of the solution of linear initial value problems for linear
operators. The recursive solution of this differential equation leads to a
peculiar Lie series, which is known as Magnus expansion, and involves Bernoulli
numbers, iterated Lie brackets and integrals. This paper aims at obtaining
further insights into the fine structure of the Magnus expansion. By using
basic combinatorics on planar rooted trees we prove a closed formula for the
Magnus expansion in the context of free dendriform algebra. From this, by using
a well-known dendriform algebra structure on the vector space generated by the
disjoint union of the symmetric groups, we derive the
Mielnik-Pleba\'nski-Strichartz formula for the continuous
Baker-Campbell-Hausdorff series.
