On the lower length of the closed-set lattice of a tree Original Research Article

Let L(T) be the closed-set latice of a tree T. The lower length l∗(L(T)) of L(T) is defined as l∗(L(T))= min≈{;|Γ|-1: Γ is a maximal chain in L(T)≈}; Call a set S of vertices in T a sparse set if d(x,y) ⩾ 3 for any two distinct vertices x, y in S. The sparsity γ(T) of T is defined as γ(T) = max ≈{;|S|: S is a sparse set of T≈}; We prove that, for any tree T of order n, l∗(L(T)) = n + 1 − γ(T) and deduce from this that l∗(L(T)) ⩾[n/2] + 1. All trees T of order n such that l∗(L(T)) = [n/2] + 1 are characterized.