Radio number of trees

A radio labeling of a graph G is a mapping f : V ( G ) ź { 0 , 1 , 2 , ź } such that | f ( u ) - f ( v ) | ź 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 ź 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.