  We discuss various forms of convergence of the vicinity of a uniformly at
random selected vertex in random simply generated trees, as the size tends to
infinity. For the standard case of a critical Galton-Watson tree conditioned to
be large the limit is the invariant random sin-tree constructed by Aldous
(1991). In the condensation regime, we describe in complete generality the
asymptotic local behaviour from a random vertex up to its first ancestor with
large degree. Beyond this distinguished ancestor, different behaviour may
occur, depending on the branching weights. In a subregime of complete
condensation, we obtain convergence toward a novel limit tree, that describes
the asymptotic shape of the vicinity of the full path from a random vertex to
the root vertex. This includes the case where the offspring distribution
follows a power law up to a factor that varies slowly at infinity.
