I think the easier relation is f(p) = p * (2 f(p) - f(p)^2) That is, a tree with an infinite path from root is a tree whose root is green, and at least one of whose children has an infinite path from root. There is an f(p)^2, yes, but no quadratic equation needed. --Michael On Sun, Mar 4, 2012 at 5:00 PM, Warren Smith <warren.wds@gmail.com> wrote:
From: Dan Asimov <dasimov@earthlink.net> I saw this problem in some stuff I was reading: Consider the standard binary tree with infinitely many levels. Suppose each edge is colored green with probability = p. What is the probability f(p) that there exists an infinite green path starting at the root?
--Let q=1-p and g(p)=1-f(p). Then g(p) obeys
g(p) = q^2 + p*q*2*g(p) + p^2 * g(p)^2
you may now solve the quadratic equation to deduce g(p) and hence f(p). Don't choose the wrong root.
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