In this work, we study asymptotics of the genealogy of
Galton--Watson
processes conditioned on the total progeny. We consider a fixed,
aperiodic and critical offspring distribution such that the rescaled
Galton--Watson processes converges to a continuous-state branching
process (CSBP) with a stable branching mechanism of index $\alpha \in (1, 2]$. We code the genealogy by two different processes: the contour
process and the height process that Le Gall and Le Jan recently
introduced. We show that the rescaled height process
of the corresponding Galton--Watson family tree, with one ancestor and
conditioned on the total progeny, converges in a functional sense, to a
new process: the normalized excursion of the continuous height process
associated with the $\alpha $-stable CSBP. We deduce from this
convergence an analogous limit theorem for the contour process. In the
Brownian case $\alpha =2$, the limiting process is the normalized
Brownian excursion that codes the continuum random tree: the result is
due to Aldous who used a different method.