Components of Random Forests

Abstract

A forest ℱ(n, M) chosen uniformly from the family of all labelled unrooted forests with n vertices and M edges is studied. We show that, like the Érdős-Rényi random graph G(n, M), the random forest exhibits three modes of asymptotic behaviour: subcritical, nearcritical and supercritical, with the phase transition at the point M = n/2. For each of the phases, we determine the limit distribution of the size of the k-th largest component of ℱ(n, M). The similarity to the random graph is far from being complete. For instance, in the supercritical phase, the giant tree in ℱ(n, M) grows roughly two times slower than the largest component of G(n, M) and the second largest tree in ℱ(n, M) is of the order n⅔ for every M = n/2 +s, provided that s3n−2 → ∞ and s = o(n), while its counterpart in G(n, M) is of the order n2s−2 log(s3n−2) ≪ n⅔.