Cutting down trees with a Markov chainsaw

Suppose you're given a tree and a chainsaw. You cut edges at random, each time keeping the portion of the tree containing the root. How many times is it necessary to cut an edge in order for the tree to be reduced to the root? We are interested in studying this process when the initial tree is random, say a uniform random labelled tree for instance. We will give a bijective proof of the asymptotic distribution of the number of cuts, hence short-cutting the current proofs of this result. The road to explaining the bijection and the main results is a great opportunity introduce many beautiful combinatorial processes on trees and lattice paths and the continuous counterpart for continuum random trees (CRT) and Brownian excursions. In particular, we will talk about the Markov chain sampling of random trees, the birthday paradox, the stick-breacking construction of the CRT, forest fragmentation processes associated to the additive coalescent, hashing with linear probing, and Poisson logging of the continuum random tree.