Minimum augmentation of a tree to a K-edge-connected graph

This paper solves the minimum augmentation problem for a given tree and positive integer k, that is, to make a tree k-edge-connected by adding the minimum number of edges. It is shown that the minimum number of edges is the least integer not less than a half of the deficiency of the tree which is defined as the sum of k-(degree) over all the vertices whose degrees are less than k. The proof is constructive and gives a polynomial-time algorithm for constructing such an augmentation.