  Consider a branching process $\{Z_n\}_{n\ge 0}$ with immigration in varying
environment. For $a\in\{0,1,2,...\},$ let $C=\{n\ge0:Z_n=a\}$ be the collection
of times at which the population size of the process attains level $a.$ We give
a criterion to determine whether the set $C$ is finite or not. For critical
Galton-Watson process, we show that $|C\cap [1,n]|/\log n\rightarrow S$ in
distribution, where $S$ is an exponentially distributed random variable with
$P(S>t)=e^{-t},\ t>0.$
