We consider a marking procedure of the vertices of a tree where each vertex is marked independently from the others with a probability that depends only on its out-degree. We prove that a critical Galton-Watson tree conditioned on having a large number of marked vertices converges in distribution to the associated size-biased tree. We then apply this result to give the limit in distribution of a critical Galton-Watson tree conditioned on having a large number of protected nodes.
['Romain Abraham', 'Aymen Bouaziz', 'Jean-François Delmas']
Local limits of galton-watson trees conditioned on the number of protected nodes