Path coverings of the vertices of a tree

Consider a collection of disjoint paths in graph G such that every vertex is on one of these paths. The size of the smallest such collection is denoted i(G). A procedure for forming such collections is established. Restricting attention to trees, the range of values for the sizes of the collections obtained is examined, and a constructive characterization of trees T for which one always obtains a collection of size i(T) is presented.