  For a complex number $\alpha$, we consider the sum of the $\alpha$th powers
of subtree sizes in Galton--Watson trees conditioned to be of size $n$.
Limiting distributions of this functional $X_n(\alpha)$ have been determined
for $\Re\alpha \neq 0$, revealing a transition between a complex normal
limiting distribution for $\Re\alpha < 0$ and a non-normal limiting
distribution for $\Re\alpha > 0$. In this paper, we complete the picture by
proving a normal limiting distribution, along with moment convergence, in the
missing case $\Re\alpha = 0$. 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.
Additionally, we prove convergence of all moments in the case $\Re\alpha < 0$,
where this result was previously missing, and establish new results about the
asymptotic mean for real $\alpha < 1/2$.
  A novel feature for $\Re\alpha=0$ is that we find joint convergence for
several $\alpha$ 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 a
logarithmic factor in the asymptotic variance when $\Re\alpha=0$; this holds
also for the shape functional.
  The proofs are largely based on singularity analysis of generating functions.
