bounding the rainbow domination number of a tree in terms of its annihilation number

a {em 2-rainbow dominating function} (2rdf) of a graph $g$ is a function $f$ from the vertex set $v(g)$ to the set of all subsets of the set ${1,2}$ such that for any vertex $vin v(g)$ with $f(v)=emptyset$ the condition $bigcup_{uin n(v)}f(u)={1,2}$ is fulfilled, where $n(v)$ is the open neighborhood of $v$. the {em weight} of a 2rdf $f$ is the value $omega(f)=sum_{vin v}|f (v)|$. the {em $2$-rainbow domination number} of a graph $g$, denoted by $gamma_{r2}(g)$, is the minimum weight of a 2rdf of g. the {em annihilation number} $a(g)$ is the largest integer $k$ such that the sum of the first $k$ terms of the non-decreasing degree sequence of $g$ is at most the number of edges in $g$. in this paper, we prove that for any tree $t$ with at least two vertices, $gamma_{r2}(t)le a(t)+1$.