  In this paper, we consider Galton-Watson processes with immigration. Pick
$i(\ge2)$ individuals randomly without replacement from the $n$-th generation
and trace their lines of descent back in time till they coalesce into $1$
individual in a certain generation, which we denote by $X_{i,1}^n$ and is
called the coalescence time. Firstly, we give the probability distribution of
$X_{i,1}^n$ in terms of the probability generating functions of both the
offspring distribution and the immigration law. Then by studying the limit
behaviors of various functionals of the Galton-Watson process with immigration,
we find the limit distribution of $X_{2,1}^n$ as $n\rightarrow\infty.$

Probability