  We consider a biased random walk $X_n$ on a Galton-Watson tree with leaves in
the sub-ballistic regime. We prove that there exists an explicit constant
$\gamma= \gamma(\beta) \in (0,1)$, depending on the bias $\beta$, such that
$X_n$ is of order $n^{\gamma}$. Denoting $\Delta_n$ the hitting time of level
$n$, we prove that $\Delta_n/n^{1/\gamma}$ is tight. Moreover we show that
$\Delta_n/n^{1/\gamma}$ does not converge in law (at least for large values of
$\beta$). We prove that along the sequences $n_{\lambda}(k)=\lfloor \lambda
\beta^{\gamma k}\rfloor$, $\Delta_n/n^{1/\gamma}$ converges to certain
infinitely divisible laws. Key tools for the proof are the classical Harris
decomposition for Galton-Watson trees, a new variant of regeneration times and
the careful analysis of triangular arrays of i.i.d. heavy-tailed random
variables.
