We compute almost surely (simultaneaously) the Hausdorff dimensions of the sets of infinite branches of the boundary of a super-critical Galton-Watson tree (endowed with a random metric) along which the averages of a vector valued branching random walk have a given set of limit points. This goes beyond multifractal analysis, for which we complete the previous works on the subject by considering the sets associated with levels in the boundary of the domain of study. Our method is inspired by some approach used to solve similar questions in the different context of hyperbolic dynamics for the Birkhoff averages of continuous potentials. It also exploits ideas from multiplicative chaos and percolation theories, which are used to estimate the lower Hausdorff dimension of a family of inhomogeneous Mandelbrot measures. This method also makes it possible to strengthen the multifractal analysis of the branching random walk averages by refining the level sets so that they contain branches over which a quantified version of the Erdös Renyi law of large numbers holds, and yields a $0-\infty$ law for the Hausdorff measures of these sets. Our results naturally give geometric and large deviations information on the heterogeneity of the birth process along different infinite branches of the Galton-Watson tree.
