α-labelings and the structure of trees with nonzero α-deficit

We present theoretical and computational results on α-labelings of trees. The theorems proved in this paper were inspired by the results of a computer investigation of α-labelings of all trees with up to 26 vertices, all trees with maximum degree 3 and up to 36 vertices, all trees with maximum degree 4 and up to 32 vertices and all trees with maximum degree 5 and up to 31 vertices. We generalise a criterion for trees to have nonzero α-deficit, and prove an unexpected result on the α-deficit of trees with a vertex of large degree compared to the order of the tree.