  In this paper, we show that a Galton-Watson tree conditioned to have a fixed
number of particles in generation $n$ converges in distribution as
$n\rightarrow\infty$, and with this tool we study the span and gap statistics
of a branching random walk on such trees, which is the discrete version of
Ramola, Majumdar and Schehr 2015, generalized to arbitrary offspring and
displacement distributions with moment constraints.
