whose ancestral line passes through $v^{(i)}$, and letting \[W\_n\^(i)(v)-def\] $W_n^{(i)} (v) := \mu^{-n} Z_n^{(i)} (v)$. Thus, for every $v$, $W(v) = \sum_i W^{(i)} (v)$. For convenience, define $p_c := 1/\mu$ \[p\_c-def\] and recall that $p_c$ is almost surely the critical percolation parameter for $\TT$ [@lyons90].

Bernoulli and Invasion Percolation
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In this subsection we give the formal construction of percolation on random trees, and for invasion percolation. Our approach is to define a simultaneous coupling of invasion percolations on all subtrees $T$ of $\ulam$ via the $U$ variables, then specialize to the random tree $\TT$. Let \[T-def\] $\T := \sigma (\{ \deg_v : v \in \VU \})$ denote the $\sigma$-field generated by the tree variables. Because $\T$ is independent from the $U$ variables, this means we have constructed a process whose law, conditional on $\T$, is invasion percolation on $\TT$. We use the notation $\E_*$\[E\_\*-def\] to denote $\E [\cdot \| \T]$; similarly $\P_* [\cdot] := \P[\cdot \| \T]$ \[P\_\*-def\]. Moreover, we use $\GW:= \P|_{\T}$ \[GW-def\] to denote the Galton-Watson measure on trees.

We begin with a similar construction for ordinary percolation. For $0 < p < 1$, simultaneously define Bernoulli$(p)$ percolations on rooted subtrees $T$ of $\ulam$ by taking the 