  A dominating broadcast on a graph G with vertex set V is a function f that
maps V to {0,1,...,diam(G)} such that f(v) does not exceed e(v) (the
eccentricity of v) for all vertices v, and each vertex u is at distance at most
f(v) from a vertex v with positive f(v). The upper broadcast domination number
of G is {\Gamma}_{b}(G), which equals the maximum of the sum of the function
values f(v), the maximum being taken over all minimal dominating broadcasts f
on G. As shown by Erwin in [D. Erwin, Cost domination in graphs, Doctoral
dissertation, Western Michigan University, 2001], {\Gamma}_{b}(G) is bounded
below by diam(G) for any graph G. We investigate trees whose upper broadcast
domination number equal their diameter and, among more general results,
characterize caterpillars with this property.
