  A branching random walk in presence of an absorbing wall moving at a constant
velocity $v$ undergoes a phase transition as the velocity $v$ of the wall
varies. Below the critical velocity $v_c$, the population has a non-zero
survival probability and when the population survives its size grows
exponentially. We investigate the histories of the population conditioned on
having a single survivor at some final time $T$. We study the quasi-stationary
regime for $v<v_c$ when $T$ is large. To do so, one can construct a modified
stochastic process which is equivalent to the original process conditioned on
having a single survivor at final time $T$. We then use this construction to
show that the properties of the quasi-stationary regime are universal when
$v\to v_c$. We also solve exactly a simple version of the problem, the
exponential model, for which the study of the quasi-stationary regime can be
reduced to the analysis of a single one-dimensional map.
