  Disordered elastic interfaces display avalanche dynamics at the depinning
transition. For short-range interactions, avalanches correspond to compact
reorganizations of the interface well described by the depinning theory. For
long-range elasticity, an avalanche is a collection of spatially disconnected
clusters. In this paper we determine the scaling properties of the clusters and
relate them to the roughness exponent of the interface. The key observation of
our analysis is the identification of a Bienaym{\'e}-Galton-Watson process
describing the statistics of the cluster number. Our work has a concrete
importance for experimental applications where the cluster statistics is a key
probe of avalanche dynamics.
