Reduced critical processes for small populations

Abstract: Let $\left{ Z(n),n\geq 1\right} $ be a critical Galton-Watson branching process with finite variance for the offspring size of particles. Assuming that $0<Z(n)\leq \varphi (n)$, where either $\varphi (n)=an$ for some $a\>0$ or $\varphi (n)=o(n)$ as $n\rightarrow \infty $, we study the structure of the process $% \left{ Z(m,n),0\leq m\leq n\right} ,$ where $Z(m,n)$ is the number of particles in the process at moment $m\leq n$ having a positive number of descendants at moment $n$.