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.
Fluctuations for the number of records on subtrees of the Continuum Random Tree