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We consider a marking procedure of the vertices of a tree where each vertexis marked independently from the others with a probability that depends only onits out-degree. We prove that a critical Galton-Watson tree conditioned onhaving a large number of marked vertices converges in distribution to theassociated size-biased tree. We then apply this result to give the limit indistribution of a critical Galton-Watson tree conditioned on having a largenumber of protected nodes.

Local limits of galton-watson trees conditioned on the number of protected nodes