What is $\operatorname{Pr}\{X_j=0|X_i=k\}$ Suppose $u_n=\operatorname{Pr}\{X_n=0|X_0=1\}$ What is $\operatorname{Pr}\{X_j=0|X_i=k\}$, where $\{X_n\}$ is a branching process and $k\geq 0$, if we were to write the answer in terms of the aforementioned $u_n$'s?

If $X_i=k$, you can also view this as $k$ different branching processes all with $X_i=1$. Then, for the total branching process to be 0 at time $j$, all $k$ branching processes have to go extinct. Then, change time by setting $t'=t-i$, and you can apply the $u_n$'s to get your answer. 
