Packing in trees

Let G be a graph and let v be a vertex of G. The open neighbourhood N(V) of v is the set of all vertices adjacent with v in G, while the closed neighbourhood of v is N(v) ∪ {v}. A packing of a graph G is a set of vertices whose closed neighbourhoods are pairwise disjoint. Equivalently, a packing of a graph G is a set of vertices whose elements are pairwise at distance at least 3 apart in G. The lower packing number of G, denoted ρ_L(G), is the minimum cardinality of a maximal packing of G while the (upper) packing number of G, denoted ρ(G), is the maximum cardinality among all packings of G. An open packing of G is a set of vertices whose open neighbourhoods are pairwise disjoint. The lower open packing number of G, denoted ρ^o_L(G), is the minimum cardinality of a maximal open packing of G while the (upper) open packing number of G, denoted ρ^o(G), is the maximum cardinality among all open packings of G. We present upper bounds on the packing number and the lower packing number of a tree. Bounds relating the packing numbers and open packing numbers of a tree are established.