  In this paper, we study a parameter that is a relaxation of arguably the most
important domination parameter, namely the domination number. Given the sheer
scale of modern networks, many existing domination type structures are
expensive to implement. Variations on the theme of dominating and total
dominating sets studied to date tend to focus on adding restrictions which in
turn raises their implementation costs. As an alternative route a relaxation of
the domination number, called disjunctive domination, was proposed and studied
by Goddard et al. A set $D$ of vertices in $G$ is a disjunctive dominating set
in $G$ if every vertex not in $D$ is adjacent to a vertex of $D$ or has at
least two vertices in $D$ at distance $2$ from it in $G$. The disjunctive
domination number, $\gamma^{d}_2(G)$, of $G$ is the minimum cardinality of a
disjunctive dominating set in $G$. We show that if $T$ is a tree of order $n$
with $l$ leaves and $s$ support vertices, then $\frac{n-l+3}{4}\leq
\gamma^{d}_2(T)\leq \frac{n+l+s}{4}$. Moreover, we characterize the families of
trees which attain these bounds.
