  Consider the edge-deletion process in which the edges of some finite tree T
are removed one after the other in the uniform random order. Roughly speaking,
the cut-tree then describes the genealogy of connected components appearing in
this edge-deletion process. Our main result shows that after a proper
rescaling, the cut-tree of a critical Galton-Watson tree with finite variance
and conditioned to have size n, converges as $n\to\infty$ to a Brownian
continuum random tree (CRT) in the weak sense induced by the Gromov-Prokhorov
topology. This yields a multi-dimensional extension of a limit theorem due to
Janson [Random Structures Algorithms 29 (2006) 139-179] for the number of
random cuts needed to isolate the root in Galton-Watson trees conditioned by
their sizes, and also generalizes a recent result [Ann. Inst. Henri
Poincar\'{e} Probab. Stat. (2012) 48 909-921] obtained in the special case of
Cayley trees.
