  A critical branching process $\left\{Z_{k},k=0,1,2,...\right\} $ in a random
environment generated by a sequence of independent and identically distributed
random reproduction laws is considered.\ Let $Z_{p,n}$ be the number of
particles at time $p\leq n$ having a positive offspring number at time $n$. \ A
theorem is proved describing the limiting behavior, as $% n\rightarrow \infty $
of the distribution of a properly scaled process $\log Z_{p,n}$ under the
assumptions $Z_{n}>0$ and $p\ll n$.
