  It has been claimed in Aldous, Miermont and Pitman [PTRF, 2004] that all
L\'evy trees are mixings of inhomogeneous continuum random trees. We give a
rigorous proof of this claim in the case of a stable branching mechanism,
relying on a new procedure for recovering the tree distance from the graphical
spanning trees that works simultaneously for stable trees and inhomogeneous
continuum random trees.
