We study the asymptotics of a ''stretched'' model of unlabeled rooted
planar trees, in which trees are not taken equiprobable but are weight
ed exponentially, according to their height. By using standard methods
for computing the probabilities of large deviations of random process
es, we show that, as the number of vertices tends to infinity, the nor
malized shape of a random tree converges in distribution to a determin
istic limit. We compute this limit explicitly. (C) 1995 John Wiley and
Sons, Inc.