  We consider a supercritical general branching population where the lifetimes
of individuals are i.i.d. with arbitrary distribution and each individual gives
birth to new individuals at Poisson times independently from each others. The
population counting process of such population is a known as binary homogeneous
Crump-Jargers-Mode process. It is known that such processes converges almost
surely when correctly renormalized. In this paper, we study the error of this
convergence. To this end, we use classical renewal theory and recent works [17,
6, 5] on this model to obtain the moments of the error. Then, we can precisely
study the asymptotic behaviour of these moments thanks to L{\'e}vy processes
theory. These results in conjunction with a new decomposition of the splitting
trees allow us to obtain a central limit theorem. MSC 2000 subject
classifications: Primary 60J80; secondary 92D10, 60J85, 60G51, 60K15, 60F05.
