A note on the maximal outdegrees of Galton–Watson trees

In this note we consider both the local maximal outdegrees and the global maximal outdegree of Galton–Watson trees. In particular, we show that the tail of any local maximal outdegree and that of the offspring distribution are asymptotically equivalent up to a certain factor. However for the global maximal outdegree, this is only true in the subcritical case.