On the minimum vertex k-path cover of trees

Let G = (V, E) and k >= 2. A subset S of V is called a vertex k-path cover if every path of order k in G contains at least one vertex from S. Minimum cardinality of a vertex k-path cover in G is denoted by psi(k) (G) and called the vertex k-path cover number of G. We find the lower bound for psi(k) in terms on the number of vertices and end vertices and characterize the extremal trees for this lower bound. In [2] it was shown that for any tree T with n vertices, psi(k)(T) <= n/k. Here we characterize extremal psi(k)-trees for this upper bound.