Modelling 2 object situation with Propositional Logic
Let $P_1$ indicate that Tree 1 has leaves (i.e. $P_1$ is true if Tree 1 has leaves), and $P_2$ indicate that Tree 2 has leaves. Sign 1 states $S_1 = P_1 \land \lnot P_2$, and Sign 2 states $S_2 = (P_1 \land \lnot P_2) \lor (\lnot P_1 \land P_2)$. We are given that $R = (S_1 \land \lnot S_2) \lor (\lnot S_1 \land S_2)$.

In order to solve this, we simply calculate $S_1,S_2$ for each of the possible truth assignments of $P_1,P_2$:
$$
\begin{array}{cc|cc|c}
P_1 &amp; P_2 &amp; S_1 &amp; S_2 &amp; R \\\hline
F &amp; F &amp; F &amp; F &amp; F \\
F &amp; T &amp; F &amp; T &amp; T \\
T &amp; F &amp; T &amp; T &amp; F \\
T &amp; T &amp; F &amp; F &amp; F
\end{array}
$$
The only truth assignment for which $R$ holds is $P_1 = F$, $P_2 = T$. We conclude that Tree 1 has no leaves and Tree 2 does have leaves.