Given a set of points in the Euclidean plane, the Euclidean \textit{$δ$-minimum spanning tree} ($δ$-MST) problem is the problem of finding a spanning tree with maximum degree no more than $δ$ for the set of points such the sum of the total length of its edges is minimum. Similarly, the Euclidean \textit{$δ$-minimum bottleneck spanning tree} ($δ$-MBST) problem, is the problem of finding a degree-bounded spanning tree for a set of points in the plane such that the length of the longest edge is minimum. When $δ\leq 4$, these two problems may yield disjoint sets of optimal solutions for the same set of points. In this paper, we perform computational experiments to compare the accuracies of a variety of heuristic and approximation algorithms for both these problems. We develop heuristics for these problems and compare them with existing algorithms. We also describe a new type of edge swap algorithm for these problems that outperforms all the algorithms we tested.