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I actually still have one question. I don't quite see how to use (2) and that $\prod_{j=2}^\infty (1+\frac{h}{j-1})=\infty$ to follow that (3) cannot hold forever. I see that using a telescopic product we can use $\prod_{j=2}^\infty (1+\frac{h}{j-1})=\infty$ to get that $c_n\to\infty$ if $m=m_n\geq n$ would not exist. But I don't know what to do next, or why this is a problem
@MathRoc Thanks for your reply! Unfortunately I only get $\theta_{n+1}=p_c\theta_n-p_c^2\theta_n^2+o(\theta_n^2)$. Its a bit hard to completely follow your proposition, since I don't speak Chinese. Does it have a name or a proof? It seems like a bit overkill, since the statement should be "not hard to see".
You are right, $r$ is the degree of the tree, constant and should be $r>2$. Otherwise our tree would be just a straight line and considering $p_c=1$ would be trivial.
