Subcritical Galton-Watson Trees
نویسنده
چکیده
Each variable X is a new, independent Uniform [0, 1] random number. For example, T = ∅ with probability 1−p, T = (∅, ∅) with probability p(1−p)2, and T = ((∅, ∅), ∅) with probability p2(1− p). The number of vertices N is equal to twice the number of left parentheses (parents) in the expression for T , plus one. Equivalently, N is twice the number of ∅s (leaves), minus one. It can be shown that N is finite with probability 1 if p ≤ 1/2 and 1/p− 1 if p > 1/2. We will focus on the subcritical case p < 1/2 for the remainder of this essay. Let Nk denote the number of vertices at distance k from the root, that is, the size of the k generation. Clearly N0 = 1 and N < ∞ if and only if Nk = 0 for all sufficiently large k. Define
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