Penalization of Galton–Watson Trees with Marked Vertices

We consider a Galton–Watson tree where each node is marked independently of each other with a probability depending on its out-degree. Using a penalization method, we exhibit new martingales where the number of marks up to level \(n-1\) $n-1$ 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.