We study the additive functional $X_n(α)$ on conditioned Galton-Watson trees given, for arbitrary complex $α$, by summing the $α$th power of all subtree sizes. Allowing complex $α$ is advantageous, even for the study of real $α$, since it allows us to use powerful results from the theory of analytic functions in the proofs. For $\Reα< 0$, we prove that $X_n(α)$, suitably normalized, has a complex normal limiting distribution; moreover, as processes in $α$, the weak convergence holds in the space of analytic functions in the left half-plane. We establish, and prove similar process-convergence extensions of, limiting distribution results for $α$ in various regions of the complex plane. We focus mainly on the case where $\Reα> 0$, for which $X_n(α)$, suitably normalized, has a limiting distribution that is not normal but does not depend on the offspring distribution $ξ$ of the conditioned Galton-Watson tree, assuming only that $E[ξ] = 1$ and $0 < \mathrm{Var} [ξ] < \infty$. Under a weak extra moment assumption on $ξ$, we prove that the convergence extends to moments, ordinary and absolute and mixed, of all orders. At least when $\Reα> \frac12$, the limit random variable $Y(α)$ can be expressed as a function of a normalized Brownian excursion.