Construction of natural cycletrees

In this article, the following question is answered: Given a cycle C, of size N, how can one compute, in O(N) time, a minimal set of edges E such that adjoining E to the cycle yields a graph with a binary spanning tree having minimal total path length? The answer is given through an algorithm for top-down construction of natural cycletrees, where the structure of each subtree is restricted by a split relation between certain properties of the subtree.