while the variables $\{ U_v \}$ will be used later for percolation. Let $\TT$\[TT-def\] be the random rooted subtree of $\ulam$ which is the connected component containing the root of the set of vertices that are either the root or are of the form $v \sqcup j$ such that $0 \leq j < \deg_v$. This is a Galton-Watson tree with ordinary generating function $\phi$.

As is usual for Galton-Watson branching processes, we denote $Z_n := |\TT_n|$. \[Z\_n-def\] Extend this by letting $Z_n (v)$ \[Z\_n(v)-def\] denote the number of offspring of $v$ in generation $|v| + n$; similarly, extend the notation for the usual martingale \[W\_n-def\] $W_n := \mu^{-n} Z_n$ by letting $W_n (v) := \mu^{-n} Z_n (v)$. \[W\_n(v)-def\] We know that $W_n (v) \to W(v)$ \[W-def\] for all $v$, almost surely and in $L^p$ if the offspring distribution has $p$ moments. This is stated without proof for integer values of $q \geq 2$ in [@harris-BP p. 16] and [@athreya-ney p. 33, Remark 3]; for a proof for all $q > 1$, see [@bingham-doney74 Theorems 0 and 5]. Further extend this notation by letting $v^{(i)}$ denote the $i$th child of $v$, letting $Z_n^{(i)} (v)$ \[Z\_n\^(i)-def\] denote $n$th generation descendants of $v$ 