  In this work, we study asymptotics of multitype Galton-Watson trees with
finitely many types. We consider critical and irreducible offspring
distributions such that they belong to the domain of attraction of a stable
law, where the stability indices may differ. We show that after a proper
rescaling, their corresponding height process converges to the continuous-time
height process associated with a strictly stable spectrally positive L\'evy
process. This gives an analogue of a result obtained by Miermont in the case of
multitype Galton-Watson trees with finite covariance matrices of the offspring
distribution. Our approach relies on a remarkable decomposition for multitype
trees into monotype trees introduced by Miermont.
