In this work, we give a description of all σ-finite measures on the space of rooted compact ℝ-trees which satisfy a certain regenerative property. We show that any infinite measure which satisfies the regenerative property is the “law” of a Lévy 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.
"Regenerative real trees." Ann. Probab. 35 (6) 2091 - 2121, November 2007. https://doi.org/10.1214/009117907000000187