  We are interested in the dynamic of a structured branching population where
the trait of each individual moves according to a Markov process. The rate of
division of each individual is a function of its trait and when a branching
event occurs, the trait of the descendants at birth depends on the trait of the
mother and on the number of descendants. In this article, we explicitly
describe the penalized Markov process, named auxiliary process, corresponding
to the dynamic of the trait along the spine by giving its associated
infinitesimal generator. We prove a Many-to-One formula and a Many-to-One
formula for forks. Furthermore, we prove that this auxiliary process
characterizes exactly the process of the trait of a uniformly sampled
individual in the large population approximation. We detail three examples of
growth-fragmentation models: the linear growth model, the exponential growth
model and the parasite infection model.
