The Degree-Constrained Minimum Spanning Tree (d-MST) problem attempts to find a minimum spanning tree with an added constraint that no nodes in the tree have a degree larger than a specified integer d. It is known that computing the d-MST is NP-hard for every d in the range 2 ≤ d ≤ (n -2), where n denotes the total number of nodes. Several approximate algorithms (heuristics) have been proposed in the literature. We have previously proposed three approximate algorithms, IR, TC-RNN, and TC-NNC, for solving the d-MST problem, the last two (TC-RNN and TC-NNC) take advantages of nearest neighbors and their properties. Our experimental results showed that both the TC-RNN and TC-NNC algorithms consistently produce spanning trees with a smaller weight (better quality-of-solution) than that of IR, but using slightly longer execution time. In this paper, we propose a new heap traversal technique that further improves the time efficiency of TC-RNN and TC-NNC. Our experiments using randomly generated, weighted graphs as inputs show that the TC-NNC algorithm outperforms the other two approximate algorithms in terms of the execution time and quality-of-solution.