We consider, for $t$ in the boundary of Galton-Watson tree ($\partial\mathsf T$), the covering number ${\mathsf N_n(t)}$ by cylinder of generation $n$. For a suitable set $I$ and a sequence $(s_{n,\gamma})$, we establish almost surely, and uniformly on $\gamma$, the Hausdorff and packing dimensions of the set $\{t\in\partial\mathsf T:{\mathsf N}_n(t)-nb\sim s_{n,\gamma}\}$ for $b\in I$.