  We investigate the combinatorial properties of the functional equation
$\phi[h(z)]=h(qz)$ for the conjugation of a formal diffeomorphism $\phi$ of
$\mathbb{C}$ to its linear part $z\mapsto qz$. This is done by interpreting the
functional equation in terms of symmetric functions, and then lifting it to
noncommutative symmetric functions. We describe explicitly the expansion of the
solution in terms of plane trees and prove that its expression on the ribbon
basis has coefficients in ${\mathbb N}[q]$ after clearing the denominators
$(q)_n$. We show that the conjugacy equation can be lifted to a quadratic fixed
point equation in the free triduplicial algebra on one generator. This can be
regarded as a $q$-deformation of the duplicial interpretation of the
noncommutative Lagrange inversion formula. Finally, these calculations are
interpreted in terms of the group of the operad of Stasheff polytopes, and are
related to Ecalle's arborified expansion by means of morphisms between various
Hopf algebras of trees.
