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We consider a critical Galton – Watson branching process starting with  N  particles; the number of offsprings is supposed to have the distribution  p   k  =( k  + 1) − τ  −( k  + 2) − τ  ,  k =0, 1, 2, … Limit distributions of the maximal tree size are obtained for the corresponding Galton – Watson forest with  N  trees and  n  non-root vertices as  N ,  n  → ∞,  n / N   τ   ⩾  C  &gt; 0.

Limit theorems for the maximal tree size of a Galton – Watson forest in the critical case