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We introduce a Poissonization method to study the coalescent structure ofuniform samples from branching processes. This method relies on the simpleobservation that a uniform sample of size $k$ taken from a random set withpositive Lebesgue measure may be represented as a mixture of Poisson sampleswith rate $\lambda$ and mixing measure $k \mathrm{d} \lambda/ \lambda$. Wedevelop a multitype analogue of this mixture representation, and use it tocharacterise the coalescent structure of multitype continuous-state branchingprocesses in terms of random multitype forests. Thereafter we study the smalltime asymptotics of these random forests, establishing a correspondence betweenmultitype continuous-state branching proesses and multitype$\Lambda$-coalescents.

The coalescent structure of uniform and Poisson samples from multitype branching processes