  It is well understood that a supercritical continuous-state branching process
(CSBP) is equal in law to a discrete continuous-time Galton Watson process (the
skeleton of prolific individuals) whose edges are dressed in a Poissonian way
with immigration which initiates subcritial CSBPs (non-prolific mass). Equally
well understood in the setting of CSBPs and super-processes is the notion of a
spine or immortal particle dressed in a Poissonian way with immigration which
initiates copies of the original CSBP, which emerges when conditioning the
process to survive eternally.
  In this article, we revisit these notions for CSBPs and put them in a common
framework using the language of (coupled) SDEs. In this way, we are able to
deal simultaneously with all types of CSBPs (supercritical, critical and
subcritical) as well as understanding how the backbone representation becomes,
in the sense of weak convergence, a spinal decomposition when conditioning on
survival. Our principal motivation is to prepare the way to expand the SDE
approach to the spatial setting of super-processes, where recent results have
increasingly sought the use of skeletal decompositions to transfer results from
the branching particle setting to the setting of measure valued processes.
