  We identify the fluctuations of the partition function of the continuous
random energy model on a Galton-Watson tree in the so-called weak correlation
regime. Namely, when the ``speed functions'', that describe the
time-inhomogeneous variance, lie strictly below their concave hull and satisfy
a certain weak regularity condition. We prove that the phase diagram coincides
with the one of the random energy model. However, the fluctuations are
different and depend on the slope of the covariance function at $0$ and the
final time $t$.
