  We consider a multi-type Galton-Watson branching processes, where the largest
in magnitude positive eigenvalue $\rho$ of the first moments matrix is close to
unity. Specifically, we examine the random vector representing the number of
individuals preceding the generation $n$, often referred to as the total
progeny. By conditioning on non-extinction or extinction at current time, and
properly normalizing it, we derive the asymptotic distribution for this vector.
Similar theorem is derived for the processes with immigration. The behavior of
this distribution is primarily influenced by the limit of $n(\rho-1)$ as $n$
tends to infinity and $\rho$ tends to 1.
