Branching processes with final types of particles and random trees

This paper considers a Bellman–Harris branching process whose probability generating function $f(s)$ of the number of direct descendants of particles satisfies the relation $f(s) = s + (1 - s)^{1 + \alpha } L(1 - s)$, $0 < \alpha \le 1$. Let $\tau$ be the moment of extinction of the process and let $\nu_\Delta$ be the total number of particles the number of direct descendants of each of which belongs to the set $\Delta ,\Delta \subset \{ 0,1, \ldots ,n, \ldots \}$. The paper gives conditions under which, for any $x \in ( - \infty , + \infty )$ and some scaling constants $b(N)$, a nondegenerate limit, $\lim _{N \to \infty } \mathbf{P}\{ \tau b(N) \le x\mid\nu_\Delta = N\}$, exists.