blocks of $H(x)$ that have non-empty intersection with the ball $B_{r-\ell}(x)$. Note that each block of $H^*(x)$ is also a block of $H(x)$. Each leaf-block of $H^*(x)$ will be called a *pre-leaf*. The pre-leaves play an important role in our analysis, because these are the objects that we will actually count.

We will prove the following four claims, which together imply the theorem.

\[clm:0\] Let $x\in X$. Each pre-leaf is an ancestor-block of at least one leaf-block of $H(x)$. Conversely, if a leaf-block $L$ of $H(x)$ has an ancestor-block which is a pre-leaf, then this pre-leaf is unique and we call it the *pre-leaf of* $L$.

\[clm:1\] Let $x\in X$ and let $L$ be a leaf-block of $H(x)$. Then there exists an $y\in X$ such that $L$ and $H(y)$ are joined by an edge of $G$.

\[clm:2\] Let $x,y \in X$ be distinct and let $L_1,L_2$ be two leaf-blocks of $H(x)$ that have distinct pre-leaves. Then at most one of $L_1, L_2$ is joined to $H(y)$ by an edge of $G$.

\[clm:3\] Let $x\in X$. Then $H(x)$ has at least $k\ell-1$ distinct pre-leaves.

Before proving these four claims, we first show how they imply the theorem. Let $x\in X$. By 