  We consider a discrete-time branching random walk defined on the real line,
which is assumed to be supercritical and in the boundary case. It is known that
its leftmost position of the $n$-th generation behaves asymptotically like
$\frac{3}{2}\ln n$, provided the non-extinction of the system. The main goal of
this paper, is to prove that the path from the root to the leftmost particle,
after a suitable normalizatoin, converges weakly to a Brownian excursion in
$D([0,1],\r)$.
