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We destroy a finite tree of size $n$ by cutting its edges one after the otherand in uniform random order. Informally, the associated cut-tree describes thegenealogy of the connected components created by this destruction process. Weprovide a general criterion for the convergence of the rescaled cut-tree in theGromov-Prohorov topology to an interval endowed with the Euclidean distance anda certain probability measure, when the underlying tree has a small height oforder $o(\sqrt{n})$. In particular, we consider uniform random recursive trees,binary search trees, scale-free random trees and a mixture of regular trees.This yields extensions of a result in Bertoin for the cut-tree of uniformrandom recursive trees and also allows us to generalize some results of Kubaand Panholzer on the multiple isolation of vertices. The approach relies in theclose relationship between the destruction process and Bernoulli bondpercolation, which may be useful for studying the cut-tree of other classes oftrees.

The cut-tree of large trees with small heights