Homomorphisms of Trees into a Path.

Let hom(G, H) denote the number of homomorphisms from a graph G to a graph H. In this paper we study the number of homomorphisms of trees into a path, and prove that hom(P-m, P-n) <= hom(T-m, P-n) <= hom(S-m, P-n), where T-m is any tree on m vertices, and P-m and S-m denote the path and star on m vertices, respectively. This completes the study of extremal problems concerning the number of homomorphisms between trees started in the paper Graph Homomorphisms Between Trees [Electron. J. Combin., 21 (2014), 4.9] written by the authors of the current paper.