Reduced critical processes for small populations

Let $\{Z(n),n\ge 1\}$ be a critical Galton-Watson branching process with finite variance for the offspring size of particles. Assuming that $0<Z(n)\le \phi (n)$, where either $\phi (n)=an$ for some $a>0$ or $\phi (n)=o(n)$ as $n\to \mathrm{\infty }$, we study the structure of the process $$ where $Z(m,n)$ is the number of particles in the process at moment $m\le n$ having a positive number of descendants at moment $n$.