Reversing the cut tree of the Brownian continuum random tree

Consider the Aldous–Pitman fragmentation process [Ann Probab, 26(4):1703–1726, 1998] of a Brownian continuum random tree ${\cal T}^{$\mathrm{b}\mathrm{r}$}$\mathrm{.}Theassociatedcuttreecut$({\cal T}^{$\mathrm{b}\mathrm{r}$})$, introduced by Bertoin and Miermont [Ann Appl Probab, 23:1469–1493, 2013], is defined in a measurable way from the fragmentation process, describing the genealogy of the fragmentation, and is itself distributed as a Brownian CRT. In this work, we introduce a shuffle transform, which can be considered as the reverse of the map taking ${\mathcal{T}}^{\mathrm{b}\mathrm{r}}$ to cut$({\cal T}^{$\mathrm{b}\mathrm{r}$})$.