Given a set S of n points in the plane, a minimum-diameter spanning tree(MDST) for the set might have a degree up to n-1. This might cause the degradation of the network performance because the node with high degree should handle much more requests than others relatively. Thus it is important to construct a spanning tree network with small degree and diameter. This paper presents an algorithm to construct a spanning tree for S satisfying the following four conditions: (1) the degree is controled as an input, (2) the tree diameter is no more than constant times the diameter of MDST, (3) the tree is monotone (even if arbitrary point is fixed as a root of the tree) in the sense that the Euclidean distance from the root to any node on the path to any leaf node is not decreasing, and (4) there are no crossings between edges of the tree. The monotone property will play a role as an aesthetic criterion in visualizing the tree in the plane.