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I am interested in the supercritical GW tree whose offsprings have positive probability to have no child conditioned on the event that the tree is not dead.

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    $\begingroup$ The keyword is "backbone decomposition". $\endgroup$
    – Martin Hairer
    Commented Jun 23, 2021 at 13:03

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Lyons, Russell, Robin Pemantle, and Yuval Peres. "Conceptual proofs of L log L criteria for mean behavior of branching processes." The Annals of Probability (1995): 1125-1138. https://www.jstor.org/stable/pdf/2244865.pdf?casa_token=VHoIvAHythwAAAAA:HGRlhOHlpz7_CFg71uL4Uj6Qc6t81IOgK3rl7T1HvOcRlYGVZPQ1HezY8-0DyLiwGRrfBf0j7GzOJFWqg7PvD2UoHT2oeAKbOkbHjutSRlpYTbL1SXY

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