  In a Bienaym\'{e}-Galton-Watson process for which there is a
positiveprobability for individuals of having no offspring, there is a
subtlebalance and dependence between the sterile nodes (the dead nodes or
leaves)and the prolific ones (the productive nodes) both at and up to the
currentgeneration. We explore the many facets of this problem, especially in
thecontext of an exactly solvable linear-fractional branching mechanism at
allgeneration. Eased asymptotic issues are investigated. Relation of
thisspecial branching process to skip-free to the left and simple random
walks'excursions is then investigated. Mutual statistical information on
theirshapes can be learnt from this association.
