  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.
