The forest associated with the record process on a L\'evy tree

TL;DR

This paper studies a pruning and regrafting process on Le9vy trees, showing the resulting tree's distribution matches the original and linking the number of cuts to leaf height, extending previous results.

Contribution

It generalizes a known result from Aldous's tree to Le9vy trees, demonstrating the invariance of the tree distribution under a specific pruning-regrafting procedure.

Findings

Regrafted tree has the same distribution as the original Le9vy tree.

Number of cuts needed to isolate the root equals the height of a random leaf.

Generalizes previous results from Aldous's tree to Le9vy trees.

Abstract

We perform a pruning procedure on a L\'evy tree and instead of throwing away the removed sub-tree, we regraft it on a given branch (not related to the L\'evy tree). We prove that the tree constructed by regrafting is distributed as the original L\'evy tree, generalizing a result where only Aldous's tree is considered. As a consequence, we obtain that the quantity which represents in some sense the number of cuts needed to isolate the root of the tree, is distributed as the height of a leaf picked at random in the L\'evy tree.