  We consider a null recurrent random walk $\mathbb{X}$ on a super-critical
Galton Watson marked tree $\mathbb{T}$ in the (sub-)diffusive regime. We are
interested in the asymptotic behaviour of the local time of its root at $n$,
which is the total amount of time spent by the random walk $\mathbb{X}$ on the
root of $\mathbb{T}$ up to the time $n$, and in its $n$-th return time to the
root of $\mathbb{T}$. We show that properly renormalized, this local time and
this $n$-th return time respectively converge in law to the maximum and to an
hitting time of some stable L\'evy process. This paper aims in particular to
extent the results of Y. Hu [Hu17].
