Cutting down $\mathbf p$-trees and inhomogeneous continuum random trees

We study a fragmentation of the $\mathbf p$-trees of Camarri and Pitman [Elect. J. Probab., vol. 5, pp. 1--18, 2000]. We give exact correspondences between the $\mathbf p$-trees and trees which encode the fragmentation. We then use these results to study the fragmentation of the ICRTs (scaling limits of $\mathbf p$-trees) and give distributional correspondences between the ICRT and the tree encoding the fragmentation. The theorems for the ICRT extend the ones by Bertoin and Miermont [Ann. Appl. Probab., vol. 23(4), pp. 1469--1493, 2013] about the cut tree of the Brownian continuum random tree.