  We consider Galton-Watson trees associated with a critical offspring
distribution and conditioned to have exactly $n$ vertices. These trees are
embedded in the real line by affecting spatial positions to the vertices, in
such a way that the increments of the spatial positions along edges of the tree
are independent variables distributed according to a symmetric probability
distribution on the real line. We then condition on the event that all spatial
positions are nonnegative. Under suitable assumptions on the offspring
distribution and the spatial displacements, we prove that these conditioned
spatial trees converge as $n\to\infty$, modulo an appropriate rescaling,
towards the conditioned Brownian tree that was studied in previous work.
Applications are given to asymptotics for random quadrangulations.
