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For a complex number $\alpha$, we consider the sum of the $\alpha$th powersof subtree sizes in Galton--Watson trees conditioned to be of size $n$.Limiting distributions of this functional $X_n(\alpha)$ have been determinedfor $\Re\alpha \neq 0$, revealing a transition between a complex normallimiting distribution for $\Re\alpha < 0$ and a non-normal limitingdistribution for $\Re\alpha > 0$. In this paper, we complete the picture byproving a normal limiting distribution, along with moment convergence, in themissing case $\Re\alpha = 0$. The same results are also established in the caseof the so-called shape functional $X_n'(0)$, which is the sum of the logarithmsof all subtree sizes; these results were obtained earlier in special cases.Additionally, we prove convergence of all moments in the case $\Re\alpha < 0$,where this result was previously missing, and establish new results about theasymptotic mean for real $\alpha < 1/2$.  A novel feature for $\Re\alpha=0$ is that we find joint convergence forseveral $\alpha$ to independent limits, in contrast to the cases$\Re\alpha\neq0$, where the limit is known to be a continuous function of$\alpha$. Another difference from the case $\Re\alpha\neq0$ is that there is alogarithmic factor in the asymptotic variance when $\Re\alpha=0$; this holdsalso for the shape functional.  The proofs are largely based on singularity analysis of generating functions.

Conditioned Galton-Watson trees: The shape functional, and more on the sum of powers of subtree sizes and its mean