Compatible spanning trees

Two plane geometric graphs are said to be compatible when their union is a plane geometric graph. Let S be a set of n points in the Euclidean plane in general position and let T be any given plane geometric spanning tree of S  . In this work, we study the problem of finding a second plane geometric tree T′T′ spanning S  , such that T′T′ is compatible with T and shares the minimum number of edges with T  . We prove that there is always a compatible plane geometric tree T′T′ having at most (n−3)/4(n−3)/4 edges in common with T, and that for some plane geometric trees T  , any plane tree T′T′ spanning S, compatible with T  , has at least (n−2)/5(n−2)/5 edges in common with T.