Abstract We consider a Galton-Watson tree where each node is marked independently of each others with a probability depending on its out-degree. Using a penalization method, we exhibit new martingales where the number of marks up to level $n-1$ appears. Then, we use these martingales to define new probability measures via a Girsanov transformation and describe the distribution of the random trees under these new probabilities.
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Abraham et al. (2024) studied this question.
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