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For a complex number α, we consider the sum of the αth powers of subtree sizes in Galton–Watson trees conditioned to be of size n. Limiting distributions of this functional X n (α) have been determined for Re α ≠ 0, revealing a transition between a complex normal limiting distribution for Re α < 0 and a non-normal limiting distribution for Re α > 0. In this paper, we complete the picture by proving a normal limiting distribution, along with moment convergence, in the missing case Re α = 0. The same results are also established in the case of the so-called shape functional X^′_n (0),which is the sum of the logarithms of all subtree sizes; these results were obtained earlier in special cases. In addition, we prove convergence of all moments in the case Re α < 0, where this result was previously missing, and establish new results about the asymptotic mean for real α < 1/2. A novel feature for Re α = 0 is that we find joint convergence for several α to independent limits, in contrast to the cases Re α ≠ 0, where the limit is known to bea continuous function of α. Another difference from the case Re α ≠ 0 is that there is a logarithmic factor in the asymptotic variance when Re α = 0; this holds also for the shape functional.

The proofs are largely based on singularity analysis of generating functions.