  We consider the spreading dynamics of two nested invasion clusters on an
infinite tree. This model was defined as the chase-escape model by Kordzakhia
and it admits a limit process, the birth-and-assassination process, previously
introduced by Aldous and Krebs. On both models, we prove an asymptotic
equivalent of the extinction probability near criticality. In the subcritical
regime, we give a tail bound on the total progeny of the preys before
extinction.
