Compatible 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 tree whose vertex set is S. The main problem we consider in this work consists of finding a second plane geometric tree T ′ on S, such that T ′ is compatible with T and shares with T a minimum number of edges. We prove, up to additive constants, that there is always a compatible plane geometric tree T ′ having in common with T at most n/4 edges; while for some plane geometric trees T , any other tree T ′ spanning S, compatible with T , has at least n/5 edges in common with T .