  To model the destruction of a resilient network, Cai, Holmgren, Devroye and
Skerman introduced the $k$-cut model on a random tree, as an extension to the
classic problem of cutting down random trees. Berzunza, Cai and Holmgren later
proved that the total number of cuts in the $k$-cut model to isolate the root
of a Galton--Watson tree with a finite-variance offspring law and conditioned
to have $n$ nodes, when divided by $n^{1-1/2k}$, converges in distribution to
some random variable defined on the Brownian CRT. We provide here a direct
construction of the limit random variable, relying upon the Aldous-Pitman
fragmentation process and a deterministic time change.
