a central unit sphere without any two spheres intersecting (beyond a point of tangency). The center points of these spheres have a minimum spanning tree in the form of a star, with the central point adjacent to all other points. Conversely, for any vertex of any minimum spanning tree, one can construct non-overlapping unit spheres centered at and at points two units along each of its edges, with a tangency for each neighbor of . Therefore, in -dimensional space the maximum possible degree of a vertex (the number of spanning tree edges connected to it) equals the kissing number of spheres in dimensions. Planar minimum spanning trees have degree at most six, and when a tree has degree six there is always another minimum spanning tree with maximum degree five. Three-dimensional minimum spanning trees have degree at most twelve. The only higher dimensions in which the exact value of the kissing number is known are four, eight, and 24 dimensions. For points generated at random from a given continuous distribution, the minimum spanning tree is almost surely unique. The numbers of vertices of any given degree converge, for large number of vertices, to a constant times that number of vertices. The values of these constants depend on the degree and the distribution. However, even for simple cases—such as the number of leaves for points uniformly distributed in a unit square—their precise values are not known. Empty regions For any edge of any Euclidean minimum spanning tree, the lens (or vesica piscis) formed by intersecting the two circles with as their radii cannot have any other given vertex in its interior. Put another way, if any tree has an edge whose lens contains a third point , then it is not of minimum length. For, by the geometry of the two circles, would be closer to both and than they are to each other. If edge were removed from the tree, would remain connected to one of and , but not the other. Replacing the removed edge by or (whichever of these two edges reconnects to the vertex from which it was disconnected) would produce a shorter tree. For any edge of any Euclidean minimum spanning tree, the rhombus with angles of 60° and 120°, having as its long diagonal, is disjoint from the rhombi formed analogously by all other edges. Two edges sharing an endpoint cannot have overlapping rhombi, because