Random cutting and records in deterministic and random trees

We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels. Limit theorems are given for this number, in particular when the tree is a random conditioned Galton–Watson tree. We consider both the distribution when both the tree and the cutting (or labels) are random and the case when we condition on the tree. The proofs are based on Aldous' theory of the continuum random tree.