\lesssim (\mu - \ee)^n$, then we say that $S$ is *exponentially small*. If every exponentially small set $S$ has $\nu_T(S) = 0$, then we say that the *the limit uniform and invasion measures on $T$ have the same entropy.*

\[th:small\] For $\GW$-a.e. $\TT$, the limit uniform and invasion measures on $\TT$ have the same entropy.

Condition on $\T$ and fix some exponentially small $S \subset \partial \TT$. Let $p \in (\frac{1}{\mu},\frac{1}{\mu-\ee})$; then since $p > p_c$, Proposition \[prop:infst\] implies that there exists an invaded vertex with an infinite subtree of weights less than $p$ below it. Let $v_p$ be the first such invaded vertex. Then the backbone is contained in the subtree below $v_p$ since the parental edge of $v_p$ together with edges of weight larger than $p$ separate the root from infinity. Moreover, all invaded vertices below $v_p$ have parental edge weight at most $p$, implying that the backbone is contained in the set of open edges below $v_p$ with weight at most $p$; call this subtree $C_p$. Note that the law of the subtree below $v_p$ (unconditioned on $\T$) with weight at most $p$ has the law of a Galton-Watson tree with progeny distribution $\Bin(Z,p)$ conditioned on $v_p$ 