  We point out that the mean-field theory of avalanches in the dynamics of
elastic interfaces, the so-called Brownian force model (BFM) developed recently
in non-equilibrium statistical physics, is equivalent to the so-called
super-Brownian motion (SBM) developed in probability theory, a continuum limit
of branching processes related to {\it space-embedded} Galton-Watson trees. In
particular the exact solvability property recently (re-)discovered from the
field theory in mean-field avalanches (the "instanton equation") maps onto the
so-called Dawson-Watanabe 1968 duality property. In the light of this
correspondence we compare the results obtained independently in the two fields,
and transport some of them from one field to the other. In particular, we
discuss a scaling limit of the branching Brownian motion which maps onto the
continuum field theory of mean-field avalanches
