to the equally splitting measure. However their results on backbones and pivots extend in a useful way to the Galton-Watson setting.

Conditioning on $T$, the way the invasion measure splits at $v$ depends on the whole tree. However, if one also conditions on the pivot at $v$, then the way the invasion measure splits at $v$ becomes independent of everything outside of the subtree at $v$. A similar statement is true if one conditions on the pivot of $v$ being less than or equal to a certain value; these are the Markov properties of Propositions \[pr:markov 2\] and \[pr:markov\]. The limiting behavior of these values are given in Theorem \[theorem:condist\] and Corollary \[cor:LPE\]. Further, Lemma \[lem:qt-ratio-survival\] shows that this conditioned splitting measure is close to a ratio of survival probabilities under supercritical Bernoulli percolation. The problem is thus reduced to proving estimates of the survival probabilities of Galton-Watson trees under supercritical Bernoulli percolation as in Section \[sec:bounds-E\].

The remainder of the paper is organized as follows. Section \[sec:prelim\] sets up the notation and gives some preliminary results. Some care is required to set up the probability space so that we can easily speak of the random measures $\mu_T$ and $\nu_T$, 