# Scaling limit of a branching process in a varying environment

A branching process in varying environment is a Galton-Watson tree whose offspring distribution can change at each generation. The evolution of the size of successive generations has drawn a lot of attention in recent years, both from the discrete and continuum points of view (the scaling limit being a modified Continuous State Branching Process).
We focus on the limiting genealogical structure, which is more delicate to study. In the critical case (all distributions have offspring mean 1), we show that under mild second moment assumptions on the sequence of offspring distributions, a BPVE conditioned to be large converges to the Brownian Continuum Random Tree, as in the standard Galton-Watson setting. The varying environment adds asymmetry and dependencies in many places. This requires numerous changes to the usual arguments. In particular we employ a (to our knowledge) new connexion between the Łukasiewicz path and the height process.
This is a joint work with Daniel Kious and Cécile Mailler.