but this class doesn’t cover the case of Galton-Watson trees conditioned on survival; they exploit quite a bit of symmetry that does not occur in the Galton-Watson case.

Let $x$ be the first invaded vertex with an infinite subtree below with weights less than $p$. Then after $x$ is invaded, no edges of weight larger than $p$ will be invaded. $\Cox$

\[cor:path\] There is almost surely only one infinite non-backtracking path from $\bf 0$ in $I$. Equivalently, $\TT$ is almost surely in the set of trees $T$ such that $I^T$ contains almost surely a unique infinite non-backtracking path from $\rtt$.

Suppose that there are two distinct paths to infinity in $I$; by Corollary \[cor-finite\], there exist maximal weights $M_1$ and $M_2$ along these paths after they split, $\P$-almost surely. If $M_1 > M_2$, the second infinite path would be invaded before the edge containing $M_1$. Similarly, we cannot have $M_2 > M_1$. Finally, $M_1 = M_2$ has $\P$-probability $0$, completing the proof. $\Cox$

This proof is stated for invasion percolation on regular trees in [@angel-invasion2008], but is identical for Galton-Watson trees once Corollary \[cor-finite\] is in place; the unique path guaranteed by \[cor:path\] is typically called the backbone of $I$, 