Direct Planar Tree Transformation and Counterexample

We consider the problem of planar spanning tree transformation in a two-dimensional plane. Given two planar trees T1 and T2 drawn on a set S of n points in general position in the plane, the problem is to transform T1 into T2 by a sequence of simple changes called edge-flips or just flips. A flip is an operation by which one edge e of a geometric object is removed and an edge f (f 6= e) is inserted such that the resulting object belongs to the same class as the original object. Generally, for geometric transformation, the usual technique is to rely on some ‘canonical’ object which can be obtained by making simple changes to the initial object and then doing the reverse operations that transform the canonical object to the desired object. In this paper, we present a technique for such transformation that does not rely on any canonical tree. It is shown that T1 and T2 can be transformed into each other by at most n − 1 + k flips (k ≥ 0) when S is in convex position and we also show results when S is in general position. We provide cases where the approach performs an optimal number of flips. A counterexample is given to show that if |T1 \ T2| = k then they cannot be transformed to one another by at most k flips.