  We investigate extremal statistical properties such as the maximal and the
minimal heights of randomly generated binary trees. By analyzing the master
evolution equations we show that the cumulative distribution of extremal
heights approaches a traveling wave form. The wave front in the minimal case is
governed by the small-extremal-height tail of the distribution, and conversely,
the front in the maximal case is governed by the large-extremal-height tail of
the distribution. We determine several statistical characteristics of the
extremal height distribution analytically. In particular, the expected minimal
and maximal heights grow logarithmically with the tree size, N, hmin ~ vmin ln
N, and hmax ~ vmax ln N, with vmin=0.373365 and vmax=4.31107, respectively.
Corrections to this asymptotic behavior are of order O(ln ln N).
