Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The first and second multiplicative Zagreb indices of $G$ are $\prod_1=\prod_{x\in V(G)}\operatorname{deg}(x)^2$ and $\prod_2=\prod_{xy\in E(G)}\operatorname{deg}(y)$, respectively, where $\operatorname{deg}(v)$ is the degree of the vertex $v$. Let $\mathcal{T}_n$ be the set of trees with $n$ vertices. We determine the elements of $\mathcal{T}_n$, extremal w.r.t. $\prod_1$ and $\prod_2$.