  The functional equation defining the free cumulants in free probability is
lifted successively to the noncommutative Fa\`a di Bruno algebra, and then to
the group of a free operad over Schr\"oder trees. This leads to new
combinatorial expressions, which remain valid for operator-valued free
probability. Specializations of these expressions give back Speicher's formula
in terms of noncrossing partitions, and its interpretation in terms of
characters due to Ebrahimi-Fard and Patras.
