Radio number for trees

Let G be a connected graph with diameter diam(G). The radio number for G, denoted by rn(G), is the smallest integer k such that there exists a function f : V (G) → {0, 1, 2, · · · , k} with the following satisfied for all vertices u and v: |f(u)−f(v)| ≥ diam(G)−dG(u, v)+1, where dG(u, v) is the distancee between u and v. We prove a lower bound for the radio number of trees, and characterize the trees achieving this bound. Moreover, we prove another lower bound for the radio number of spiders (trees with at most one vertex of degree more than two) and characterize the spiders achieving this bound. Our results generalize the radio number for paths obtained by Liu and Zhu.