The contour process of a random binary tree t with n internal nodes is defined as the polygonal function constructed from the heights of the leaves of t (normalized by ). We show that, as n --> [infinity], the limiting contour process is identical in distribution to a Brownian excursion.

It is well known, that under the condition LAN and some more regularity conditions, the process of log-likelihood functions converges weakly to a degenerate Gaussian process (the trajectories of which are straight lines). In the non-regular case considered by several authors [1, 9] the limiting...