percolation clusters to be the connected component containing $\rtt$ of the induced subtrees of $T$ on all vertices $v$ such that $U_v \leq p$. Let $\F_n$ \[F\_n-def\] be the $\sigma$-field generated by the variables $\{ U_v , \deg_v : |v| < n \}$. Let $p_c = 1/\mu = 1/\phi'(1)$ denote the critical probability for percolation. Write $v \conn_{T,p} w$ \[conn-def\] if $U_u \leq p$ for all $u$ on the geodesic from $v$ to $w$ in $T$. Informally, $v \conn_{T,p} w$ iff $v$ and $w$ are both in $T$ and are connected in the $p$-percolation. The event of successful $p$ percolation on $T$ is $H_T (p) := \{ \rtt \conn_{T,p} \infty \}$ and the event of successful $p$ percolation on the random tree $\TT$, is denoted $H_\TT (p)$ or simply\[H(p)-def\] $H(p)$. Let \[g(T,p)-def\] $g(T , p) := 
\P [H_T (p)]$ denote the probability of $p$ percolation on $T$. The conditional probability $\P_* [H(p)]$ is measurable with respect to $\TT$ and we may define $g(\TT , p):= \P_* [H(p)]$. Furthermore, we may define \[g(p)-def\] $g(p) = \P [H(p)] = \E g(\TT , p)$. Since $p_c = 1/\mu$ is the critical percolation parameter for a.e. $\TT$, note that $g(\TT,p) = 0$ for all 