We consider  a super-critical Galton-Watson tree  T whose non-degenerate
offspring distribution has finite mean.  We consider the random trees Tn
distributed as T conditioned on the n-th generation, Z(n), to be of size
a(n). We identify the possible local limits  of Tn as n goes to infinity
according to the growth rate of a(n). In the low regime, the local limit
is the Kesten  tree, in the moderate regime the  family of local limits,
is distributed  as T  conditionally on {W  =q}, where q>0  and W  is the
(non-trivial) limit of the renormalization  of Z(n). In the high regime,
we prove  the local convergence  towards a  limiting tree 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 (as a function of q).