Bifurcations of Steiner minimal trees and minimal fillings for non-convex four-point boundaries and Steiner subratio for the Euclidean plane

Bifurcation diagrams for topologies of Steiner minimal trees and minimal fillings for nonconvex four-point boundaries in the Euclidean plane are constructed. Using this result, the four-pointed Steiner subratio of the Euclidean plane is obtained. All configurations which it is attained at are found.