distance away from $\rho = 0$.

\[pr:KL\] Let $\rho$ and $\rho'$ be probability measures on the set $\{ 1, \ldots , k \}$ and denote $\ee_i := \rho'(i) / \rho(i) - 1$. Then $$\KL (\rho , \rho') \leq \sum_{i=1}^k \rho(i) \ee_i^2 \, .$$

Define the function $R$ on $(-1,\infty)$ by $$R(x) := \frac{(1+x) \log (1+x) - x}{x^2}$$ if $x \neq 0$ and $R(0) := 1/2$. This makes $R$ continuous, positive, and decreasing from 1 to 0 on $(-1,\infty)$. When $\ee = \rho'/\rho - 1$, we may compute $$\frac{\rho' \log (\rho'/\rho) - (\rho'-\rho)}{\rho} 
   = \frac{(1+\ee) \rho \log (1+\ee) - \ee \rho}{\rho} = \ee^2 R(\ee) \, .$$ Because $\sum_{i=1}^k \rho(i) = \sum_{i=1}^k \rho'(i) = 1$, we see that $$\KL (\rho, \rho') 
   = \sum_{i=1}^k (\rho'(i) - \rho(i)) + \rho(i) \ee_i^2 R(\ee_i) 
   = \sum_{i=1}^k \rho(i) \ee_i^2 R(\ee)$$ and the result follows from $0 < R(\ee) < 1$. $\Cox$

Applying Proposition \[pr:KL\] to $\rho' = q$ and $\rho = p$ gives $$\label{eq:KL}
    X(u) \leq \sum_{w}p(w)\ee(w)^2$$ where $\ee(w) = \frac{q(w)}{p(w)} - 1.$

\[lem:X\] Let $T$ be a fixed tree on which $\nu_T$ and $\mu_T$ are well defined on the Borel $\sigma$-field $\B$ on $\partial T$. If $$\label{eq:L1}
\sum_{n=1}^\infty \sum_{|v|=n} X(v) \nu_T (v) < \infty$$ then $\nu_T 