The process we consider is a binary splitting, where the probability of division, p//i equals one-half plus 1/2i, i equals 1,2, . . . , depends on the population size, 2i. We show that Z//n converges to infinity almost surely on set Q OVER BAR of positive probability. Z//n/N converges in distribution to a proper limit, SIGMA ** infinity //n// equals //0(1/Z//n) diverges almost surely on Q OVER BAR , SIGMA ** infinity //n// equals //0(1/Z**2//n) converges almost surely on Q OVER BAR and there are no constants C//n such that Z//n/c//n converges in probability to a non-degenerate limit.