(2,1)-Total number of trees with maximum degree three

The (2,1)-total labelling number of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that no two adjacent vertices have the same label, no two adjacent edges have the same label and the difference between the labels of a vertex and its incident edges is at least 2. It is known that every tree T with maximum degree Δ has . In this paper, we characterize completely the (2,1)-total number of trees with Δ=3.