Normal convergence problem? Two moments and a recurrence may be the clues

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.