  In this work, we give a description of all sigma-finite measures on the space
of rooted compact real trees which satisfy a certain regenerative property. We
show that any infinite measure which satisfies the regenerative property is the
"law" of a Levy tree, that is, the "law" of a tree-valued random variable that
describes the genealogy of a population evolving according to a
continuous-state branching process. On the other hand, we prove that a
probability measure with the regenerative property must be the law of the
genealogical tree associated with a continuous-time discrete-state branching
process.
