  We encode the genealogy of a continuous-state branching process associated
with a branching mechanism $\Psi$ - or $\Psi$-CSBP in short - using a
stochastic flow of partitions. This encoding holds for all branching mechanisms
and appears as a very tractable object to deal with asymptotic behaviours and
convergences. In particular we study the so-called Eve property - the existence
of an ancestor from which the entire population descends asymptotically - and
give a necessary and sufficient condition on the $\Psi$-CSBP for this property
to hold. Finally, we show that the flow of partitions unifies the lookdown
representation and the flow of subordinators when the Eve property holds.
