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Total progeny of a Galton-Watson branching process - standard textbook question
The answer above is fine, nevertheless I make some hopefully useful supplementary remarks (the first two essentially reformulating Did's answer)
(1) It is well known (see e.g. Feller I, 3rd ed., p.29 …
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Multivariate Generating Function Related to Lambert $W$ Function and Counting Trees with a C...
The asymptotic distribution of the number of nodes at maximal height in a random tree is known.
The following was known as "Wilf's conjecture" (H. S. Wilf stated it in 1991, evidently unaware of the …