Gibbs fragmentation trees

We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs type fragmentation tree with Aldous's beta-splitting model, which has an extended parameter range $\beta>-2$ with respect to the ${\rm Beta}(\beta+1,\beta+1)$ probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the two-parameter Poisson-Dirichlet models for exchangeable random partitions of $\bN$, with an extended parameter range $0\le\alpha\le 1$, $\theta\ge -2\alpha$ and $\alpha<0$, $\theta=-m\alpha$, $m\in\bN$.