Martin-Löf randomness and Galton-Watson processes

The members of Martin-Löf random closed sets under a distribution studied by Barmpalias et al. are exactly the infinite paths through Martin-Löf random Galton--Watson trees with survival parameter $\frac{2}{3}$. To be such a member, a sufficient condition is to have effective Hausdorff dimension strictly greater than $γ=\log_2 \frac{3}{2}$, and a necessary condition is to have effective Hausdorff dimension greater than or equal to $γ$.