Galton –Watson forests with N rooted trees and n nonroot vertices are considered. It is assumed that such forests are formed by a critical branching process in which the number of offsprings for each particle has the distribution
p_k = h(k + 1)/(k + 1)^τ, k=0, 1, 2, . . . , τ ∈ (2, 3), where the function h(x) is slowly varying at infinity. Some limit distributions of the number of trees of a given size are found if N, n→∞, n/N →∞.