  Let $G=(V,E)$ be a graph. A subset $D$ of $V(G)$ is called a super dominating
set if for every $v \in V(G)-D$ there exists an external private neighbour of
$v$ with respect to $V(G)-D.$ The minimum cardinality of a super dominating set
is called the super domination number of $G$ and is denoted by
$\gamma_{sp}(G)$. In this paper some results on the super domination number are
obtained. We prove that if $T$ is a tree with at least three vertices, then
$\frac{n}{2}\leq\gamma_{sp}(T)\leq n-s,$ where $s$ is the number of support
vertices in $T$ and we characterize the extremal trees.
