  Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary
tree representing the evolution of a trait along a population where each
individual has two children. We provide a central limit theorem for additive
functionals of BMC under $L^2$-ergodic conditions with three different regimes.
This completes the pointwise approach developed in a previous work. As
application, we study the elementary case of symmetric bifurcating
autoregressive process, which justify the non-trivial hypothesis considered on
the kernel transition of the BMC. We illustrate in this example the phase
transition observed in the fluctuations.
