A note on trees of maximum weight and restricted degrees

For a graph G let w−1(G) be the sum of (dG(u)dG(v))−1 over all edges uv of G. Clark and Moon (Ars Combin. 54 (2000) 223–235) proved an upper bound on w−1 for trees and posed the problem to determine a best possible such bound. In the present paper, we do this for trees of maximum degree 3. Furthermore, we prove an asymptotically best possible upper bound on w−1 for trees such that all degrees of vertices are either 1, 2 or some fixed Δ⩾4.