Radio number of trees

A radio labeling of a graph G is a mapping f:V(G)→{0,1,2,…} such that |f(u)−f(v)|≥diam(G)+1−d(u,v) for every pair of distinct vertices u,v of G, where diam(G) is the diameter of G and d(u,v) the distance between u and v in G. The radio number of G is the smallest integer k such that G has a radio labeling f with max{f(v):v∈V(G)}=k. We give a necessary and sufficient condition for a lower bound on the radio number of trees to be achieved, two other sufficient conditions for the same bound to be achieved by a tree, and an upper bound on the radio number of trees. Using these, we determine the radio number for three families of trees.