any sequence $\bfa = (a_1, \ldots , a_n)$ to any extension $\bfa \sqcup j := (a_1, \ldots , a_n , j)$. The depth of a vertex $v$ is the graph distance between $v$ and $\rt$ and is denoted $|v|$.\[depth-def\] We work with trees $T$ that are locally finite rooted subtrees of $\ulam$. The usual notations are in force: $T_n$ \[T\_n-def\] denotes the set of vertices at depth $n$; $T(v)$ \[T(v)-def\] is the subtree of $T$ at $v$, canonically identified with a rooted subtree of $\ulam$, in other words the vertex set of $T(v)$ is $\{ w : v \sqcup w \in V(T) \}$ \[sqcup-def\]; $\partial T$ \[partial-T-def\] denotes the set of infinite non-backtracking paths from the root; if $\gamma \in \partial T$ then \[gamma\_n-def\] $\gamma_n$ $(n \geq 0)$ denotes the $n$th vertex in $\gamma$; the last common ancestor of $v$ and $w$ is denoted $v \wedge w$ \[LCA-vertex-def\] and the last common vertex of $\gamma$ and $\gamma'$ is denoted $\gamma \wedge \gamma'$ \[LCA-path-def\]; $\Par{v}$ denotes the parent of $v$ \[Par-def\]. Let $\mu_T^n$ \[mu\_T\^n-def\] denote the uniform measure on the $n$th generation of $T$. In some cases, for example for almost every Galton-Watson tree, the limit $\mu_T := \lim_{n \to \infty} 