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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 aforest of such trees converges in law to the reflected Brownian motion. We thenuse this to deduce under mild conditions an invariance principle for multitypeGalton--Watson trees with a countable number of types, thus extending a resultof G. Miermont on multitype Galton--Watson trees with finitely many types.

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