A planar binary tree is a tree in the plane all of whose vertices have degree 1 or 3; minimum Steiner trees having no vertex of degree 2 are trees of that kind. For such a tree \(\Gamma\) the authors define a twisting number \(\text{tw }\Gamma\) as follows: The twisting number of a directed path in \(\Gamma\) is the difference between the number of left-hand branchings and the number of right-hand branchings along that path, and then \(\text{tw }\Gamma\) is the maximum of the twisting numbers of all directed paths. Finally, a finite point set \(M\) is said to have \(k\) convexity levels iff \(M\) is the disjoint union of \(M_1,\dots, M_k\) where \(M_j\) \((j= 1,\dots,k)\) is the set of those points of \(M_j\cup\cdots\cup M_k\) which are lying on the boundary of the convex hull of this set. Now, the authors’ main result says that the twisting number of a planar binary minimum Steiner tree spanning a set \(M\) that has \(k\) convexity levels is bounded by \(12(k- 1)+5\) and this bound is best possible. This generalizes earlier results of the authors concerning the case \(k=1\). The main part of the lengthy paper is devoted to the investigation of the twisting number by considering analoguous notions for general polygonal lines.