$(1-tx)^{-1/t} - 1$ is the e.g.f. for a plane $m$-ary tree when $t=m-1$. OEIS A094638 provides some examples when $t = -1,-2,-3$ in my Dec. 15, 2007, comments. The e.g.f. is of importance for $t$ any real number.
My comments on the relation between $[A^{(m)}]$ for $m \geq 1$ and the Fuss-Catalan numbers, generated by compositional inversion of $f(x) = x \pm x^{m+1}$ about $x=0$ (see, e.g., A001764), in my answer / extension to the MO-Q "Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution" has another perspective since $m$ is extended there to any integer. This is a generalization of the formalism of Novelli and Thibon in "Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions".
(Often the presence of negative integers in a generating function indicates some combinatorics of an underlying topological nature, such as Euler's formula for polytopes. This MO-Q contains another example of how natural it can be to extend $n$ in significant combinatorial sequences from the natural numbers to the full integers and retain combinatorial import.)