  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.
