LetGbe a graph and letvbe a vertex ofG. The open neigbourhoodN(v)ofvis the set of all vertices adjacent withvinG. An open packing ofGis a set of vertices whose open neighbourhoods are pairwise disjoint. The lower open packing number ofG, denotedρ°L(G), is the minimum cardinality of a maximal open packing ofGwhile the (upper) open packing number ofG, denotedρ°(G), is the maximum cardinality among all open packings ofG. It is known (see [7]) that ifGis a connected graph of order n ≥3, thenρ°(G)≤ 2n/3 and this bound is sharp (even for trees). As a consequence of this result, we know thatρ°L(G)≤ 2n/3. In this paper, we improve this bound whenGis a tree. We show that ifGis a tree of ordernwith radius 3, thenρ°L(G)≤n/2 + 2 √n-1, and this bound is sharp, while ifGis a tree of ordernwith radius at least 4, thenρ°L(G)is bounded above by 2n/3—O√n).