We consider a super-critical Galton-Watson tree whose non-degenerate offspring distribution has finite mean. We consider the random trees $τ$n distributed as $τ$ conditioned on the n-th generation, Zn, to be of size an $\in$ N. We identify the possible local limits of $τ$n as n goes to infinity according to the growth rate of an. In the low regime, the local limit $τ$ 0 is the Kesten tree, in the moderate regime the family of local limits, $τ$ $θ$ for $θ$ $\in$ (0, +$\infty$), is distributed as $τ$ conditionally on {W = $θ$}, where W is the (non-trivial) limit of the renormalization of Zn. In the high regime, we prove the local convergence towards $τ$ $\infty$ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits ($τ$ $θ$ , $θ$ $\in$ [0, $\infty$]).