Simulating theta random variates

We develop an exact simple random variate generator for the theta distribution, which occurs as the limit distribution of the height of nearly all models of uniform random trees. Even though the density is only known as an infinite sum of functions, our algorithm does not require any summation. The properties of the theta distribution with distribution function F(x)= ~ (1-2j2x2)e-J2x2 j~-(x3 oo-4~Z5/2x3 ~ j2e-n2J2/x2, x>0 j=l are described in Rfnyi and Szekeres (1967). The theta distribution occurs as the limit law for the height of a random rooted labeled free tree. It also is the limit law of the height of many other brands of random trees, such as random planted planar trees (or rooted ordered trees; see DeBruijn et al., 1972). As shown in Flajolet and Odlyzko (1982), this is no coincidence. In fact, for all so-called simply generated families of trees, the theta distribution describes the limit law. The density of the distribution is plotted below. Its sth moment,/Is is given by Us = 2r(1 + s/2)(s-1)((s), where ((s) =)-'~n~=l 1/n ~ is the zeta function. In particular, the mean is ~ and the variance n(~z-3)/2.