Let $\mathbb{A}$ be an admissible set. A sentence of the form $\forall\bar{R}\phi$ is a $\forall_1(\mathbb{A}) (\forall^s_1(\mathbb{A}),\forall_1(\mathscr{L}_{\omega_1\omega}))$ sentence if $\phi \in \mathbb{A} (\phi$ is $\bigvee\Phi$, where $\Phi$ is an $\mathbb{A}$-r.e. set of sentences from $\mathbb{A}; \phi \in \mathscr{L}_{\omega_1\omega}$). A sentence of the form $\exists\bar{R}\phi$ is an $\exists_2(\mathbb{A}) (\exists^s_2(\mathbb{A}),\exists_2(\mathscr{L}_{\omega_1\omega}))$ sentence if $\phi$ is a $\forall_1(\mathbb{A}) (\forall^s_1(\mathbb{A}),\forall_1(\mathscr{L}_{\omega_1\omega}))$ sentence. A class of structures is, for example, a $\forall_1(\mathbb{A})$ class if it is the class of models of a $\forall_1(\mathbb{A})$ sentence. Thus $\forall_1(\mathbb{A})$ is a class of classes of structures, and so forth. Let $\mathfrak{M}_i$ be the structure $\langle i, <\rangle$, for $i > 0$. Let $\Gamma$ be a class of classes of structures. We say that a sequence $J_1,\ldots,J_i,\ldots, i < \omega$, of classes of structures is a $\Gamma$ sequence if $J_i \in \Gamma, i < \omega$, and there is $I \in \Gamma$ such that $\mathfrak{M} \in J_i$ if and only if $\lbrack\mathfrak{M}, \mathfrak{M}_i\rbrack \in I$, where $\lbrack,\rbrack$ is the disjoint sum. A class $\Gamma$ of classes of structures has the easy uniformization property if for every $\Gamma$ sequence $J_1,\ldots,J_i, \ldots, i < \omega$, there is a $\Gamma$ sequence $J'_1,\ldots,J'_i,\ldots, i < \omega$, such that $J'_i \subseteq J_i, i < \omega, \bigcup J'_i = \bigcup J_i$, and the $J'_i$ are pairwise disjoint. The easy uniformization property is an effective version of Kuratowski's generalized reduction property that is closely related to Moschovakis's (topological) easy uniformization property. We show over countable structures that $\forall_1(\mathbb{A})$ and $\exists_2(\mathbb{A})$ have the easy uniformization property if $\mathbb{A}$ is a countable admissible set with an infinite member, that $\forall^s_1(\mathscr{L}_\alpha)$ and $\exists^s_2(\mathscr{L}_\alpha)$ have the easy uniformization property if $\alpha$ is countable, admissible, and not weakly stable, and that $\forall_1(\mathscr{L}_{\omega_1\omega})$ and $\exists_2(\mathscr{L}_{\omega_1\omega})$ have the easy uniformization property. The results proved are more general. The result for $\forall^s_1(\mathscr{L}_\alpha)$ answers a question of Vaught (1980).