  We consider branching random walks built on Galton-Watson trees with
offspring distribution having a bounded support, conditioned to have $n$ nodes,
and their rescaled convergences to the Brownian snake. We exhibit a notion of
"globally centered discrete snake'' that extends the usual settings in which
the displacements are supposed centered. We show that under some additional
moment conditions, when $n$ goes to $+\infty$, "globally centered discrete
snakes'' converge to the Brownian snake. The proof relies on a precise study of
the "lineage'' of the nodes in a Galton-Watson tree conditioned by the size,
and their links with a multinomial process. Some consequences concerning
Galton-Watson trees conditioned by the size are also derived.
