I found this example in "A Course in Model Theory", but don't seem to figure out why it is true.
Let $L$ be a language having a unary predicate $P_s$ for each (finite) binary string $s \in \{0,1\}^*$ and let $Tree$ be the theory saying that the $P_s$ form a binary decomposition of the universe. That is, for every binary string $s\in\{0,1\}^*$ the theory $Tree$ contains the sentences:
$\forall x P_{\langle\rangle}(x)$
$\exists x P_s(x)$
$\forall x ((P_{s0} (x) \vee P_{s1} (x)) \iff P_s (x))$
$\forall x \neg(P_{s0}(x) \wedge P_{s1} (x))$
Show that $Tree$ has quantifier elimination and that $Tree$ is complete.