Broadcast independence and packing in certain classes of trees

Given a graph $G=(V,E)$ of diameter d, a broadcast is a function f:V(G) → \ 0, 1, , d \ where $f(v)$ is at most the eccentricity of v. A vertex v is broadcasting if $f(v)>0$ and a vertex u hears v if d(u,v) ≤ f(v). A broadcast is independent if no broadcasting vertex hears another vertex and is a packing if no vertex hears more than one vertex. The weight of f is ∑v ∈ V f(v). We find the maximum weight independent and packing broadcasts for perfect k-ary trees, spiders, and double spiders as a partial answer to a question posed by Ahmane et al.