Broadcasts and domination in trees

A broadcast   on a graph GG is a function f:V→Z+∪{0}f:V→Z+∪{0}. The broadcast number   of GG is the minimum value of ∑v∈Vf(v)∑v∈Vf(v) among all broadcasts ff for which each vertex of GG is within distance f(v)f(v) from some vertex vv with f(v)≥1f(v)≥1. This number is bounded above by the radius and the domination number of GG. We show that to characterize trees with equal broadcast and domination numbers it is sufficient to characterize trees for which all three of these parameters coincide.