Reduced critical Bellman-Harris branching processes for small populations

Let $\{Z(t),t\ge 0\}$ be a critical Bellman-Harris branching process with finite variance for the offspring size of particles. Assuming that $0<Z(t)\le \phi (t)$, where either $\phi (t)=o(t)$ as $t\rightarrow \infty $or$\varphi (t)=at,, a>0$,westudythestructureoftheprocess$% \left{ Z(s,t),0\leq s\leq t\right} ,$where$Z(s,t)$ is the number of particles in the process at moment $s$ in the initial process which either survive up to moment $t$ or have a positive offspring number at this moment.