We consider a random forest formed by trajectories of a homogeneous Galton – Watson branching process starting with N particles, where the number of immediate offspring of each particle has the distribution pk = (k + 1)^−τ − (k + 2)^−τ, k = 0,1,2,... . For the Galton –Watson forest containing at most n vertices, we find limit distributions of the maximum tree size as N, n→∞ for different values of the parameter τ corresponding to a critical or subcritical branching process.