For various global characteristics of large size combinatorial
structures, such as graphs, trees, one can usually estimate the mean and the
variance, and also obtain a recurrence for the generating function, with the
structure size n serving as the recursive parameter. As a heuristic
principle based on our experience, we claim that such a characteristic is
asymptotically normal if the mean and the variance are "nearly linear" in
n. The technical reason is that in such a case the moment generating
function (the characteristic function) of the normal distribution with the same
two moments "almost" satisfies the recurrence. Of course, an actual proof
may well depend on a magnitude of the relative error, and the latter is
basically determined by degree of nonlinearity of the mean and the variance. We
provide some new illustrations of this paradigm. The uniformly random tree on
n-labelled vertices is studied. Using and strengthening the earlier
results of Meir and Moon, we show that the independence number is asymptoticaly
normal, with mean $\rhon$ and variance $\sigma^2n, (\sigma^2 =
\rho(1-\rho-\rho^2)(1+\rho)^{-1})$; here $\rho \approx 0.5671$ is the root of
$xe^x = 1$. As a second example, we prove that--in the rooted tree--the
counts of vertices with total progeny of various sizes form an asymptotically
Gaussian sequence. This is an extension of Rényi's result on asymptotic
normality of the number of leaves in the random tree. In both cases we
establish convergence of the generating function. Finally we show that the
overall number of ways to extend, totally, the tree-induced partial order is
lognormal in the limit, with mean and variance roughly $\logn!-an$ and $bn \log
n$. The proof is based on convergence of the cumulants of the generating
function.