This thesis concerns different skeletal decompositions of various branching processes. First we develop a stochastic differential equation (SDE) approach to describe the fitness of certain sub-populations in asexual high-density stochastic population models. Initially we only consider continuous-state branching processes, then we extend the SDE approach to the spatial setting of superprocesses. In both cases the SDE can be used to simultaneously describe the total mass, and those embedded genealogies that propagate prolific traits in surviving populations, where 'survival' can be interpreted in different ways. For example, it can mean survival beyond a certain time-horizon, but it can also mean survival according to some spatial criteria. In the second part of the thesis we construct the prolific backbone decomposition of multitype superprocesses by extending the semigroup approach already available for one-type branching processes.