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)|≥d+1−d(u,v) for every pair of distinct vertices u, v of G, where d and d(u,v) are the diameter of G and the distance between u and v in G, respectively. 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 present a lower bound for the radio number of trees and a necessary and sufficient condition for this bound to be achieved. Using this condition we determine the radio number for three families of trees.