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Abstract: We study fragmentation trees of Gibbs type. In the binary case, we identifythe most general Gibbs-type fragmentation tree with Aldous beta-splittingmodel, which has an extended parameter range $\beta>-2$ with respect to the${ m beta}\beta+1,\beta+1$ probability distributions on which it is based.In the multifurcating case, we show that Gibbs fragmentation trees areassociated with the two-parameter Poisson-Dirichlet models for exchangeablerandom partitions of $\mathbb {N}$, with an extended parameter range$0\le\alpha\le1$, $\theta\ge-2\alpha$ and $\alpha<0$, $\theta =-m\alpha$, $m\in\mathbb {N}$.

Author: Peter McCullagh, Jim Pitman, Matthias Winkel


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