Scaling limit of multitype Galton-Watson trees with infinitely many types

We introduce a certain class of 2-type Galton-Watson trees with edge lengths. We prove that, after an adequate rescaling, the weighted height function of a forest of such trees converges in law to the reflected Brownian motion. We then use this to deduce under mild conditions an invariance principle for multitype Galton–Watson trees with a countable number of types, thus extending a result of G. Miermont on multitype Galton–Watson trees with finitely many types.