  We consider the asymptotics of various estimators based on a large sample of
branching trees from a critical multi-type Galton-Watson process, as the sample
size increases to infinity. The asymptotics of additive functions of trees,
such as sizes of trees and frequencies of types within trees, a higher-order
asymptotic of the ``relative frequency'' estimator of the left eigenvector of
the mean matrix, a higher-order joint asymptotic of the maximum likelihood
estimators of the offspring probabilities and the consistency of an estimator
of the right eigenvector of the mean matrix, are established.
