  Consider a rooted $N$-ary tree. To every vertex of this tree, we attach an
i.i.d. continuous random variable. A vertex is called accessible if along its
ancestral line, the attached random variables are increasing. We keep
accessible vertices and kill all the others. For any positive constant
$\alpha$, we describe the asymptotic behaviors of the population at the $\alpha
N$-th generation as $N$ goes to infinity. We also study the criticality of the
survival probability at the $(eN-\frac{3}{2}\log N)$-th generation in this
paper.
