 Solution

The Galton-Watson extinction fixed point equation isIts smallest root in $[0,1]$ is $q=(3\sqrt{3}-5)/4$, not the other allowed fixed point $1$. Consequently the large-depth escape probability isFor finite $N$, the exact escape probability is $1-q_N$ and is slightly larger. Indeed ${\varphi }^{\prime }(q)=2-\sqrt{3}<1$, and the mean value theorem on $[q_N,q]$ gives $0\le q-q_N\le q(2-\sqrt{3}{)}^N$. This also quantifies how quickly the finite-depth answer approaches the displayed limit.