Pruning, cut trees, and the reconstruction problem

We consider a pruning of the inhomogeneous continuum random trees, as well as the cut trees that encode the genealogies of the fragmentations that come with the pruning. We propose a new approach to the reconstruction problem, which has been treated for the Brownian CRT in [Electron. J. Probab. vol. 22, 2017] and for the stable trees in [Ann. IHP B, vol 55, 2019]. Our approach does not rely upon self-similarity and can potentially apply to general L\'evy trees as well.