Explore the individuals one at a time, maintaining the number of discovered but unprocessed individuals. It starts at . Processing one individual removes it and adds its offspring, so
(1)
The offspring choices make these increments independent. Exploration stops when the queue first becomes empty; every processed individual has then been counted exactly once. Hence
(2)
with if the walk never hits zero. The increment sequence can be extended independently beyond the stopping time for probability estimates.

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