  We study the asymptotic behavior af the number of cuts $X(T_n)$ needed to
isolate the root in a rooted binary random tree $T_n$ with $n$ leaves. We focus
on the case of subtrees of the Continuum Random Tree generated by uniform
sampling of leaves. We elaborate on a recent result by Abraham and Delmas, who
showed that $X(T_n)/\sqrt{2n}$ converges a.s. towards a Rayleigh-distributed
random variable $\Theta$, which gives a continuous analog to an earlier result
by Janson on conditioned, finite-variance Galton-Watson trees. We prove a
convergence in distribution of $n^{-1/4}(X(T_n)-\sqrt{2n}\Theta)$ towards a
random mixture of Gaussian variables. The proofs use martingale limit theory
for random processes defined on the CRT, related to the theory of records of
Poisson point processes.
