  We consider a branching particle system consisting of particles moving
according to the Ornstein-Uhlenbeck process in $\Rd$ and undergoing a binary,
supercritical branching with a constant rate $\lambda>0$. This system is known
to fulfil a law of large numbers (under exponential scaling). Recently the
question of the corresponding central limit theorem has been addressed. It
turns out that the normalization and form of the limit in the CLT fall into
three qualitatively different regimes, depending on the relation between the
branching intensity and the parameters of the Orstein-Uhlenbeck process. In the
present paper we extend those results to $U$-statistics of the system proving a
law of large numbers and a central limit theorem.
