We consider, for t in the boundary of a Galton–Watson tree $(\partial \textsf{T})$ , the covering number $(\textsf{N}_n(t))$ by the generation-n cylinder. For a suitable set I and sequence (s_n), we almost surely establish the Hausdorff dimension of the set $\{ t \in \partial {\textsf{T}}:{{\textsf{N}}_n}(t) - nb \ {\sim} \ {s_n}\} $ for b ∈ I.