Let T and T' be two unrooted trees. In this paper we will explicitly refer to the leaves of a tree as leaves and the non-leaf nodes as internal node. We will assume that T and T' each has n labelled leaves numbered 1,..., n such that the leaf numbered x in T has the same label as the leaf numbered x in T'. The leaf sets are denoted L and L' for T and T' respectively, note that L = L'. We will use V and V' to denote the internal nodes in T and T' respectively. The degree of an internal node v is the number of subtrees connected to it, and is denoted dv. The internal degree of an internal node v, idv, is the number of non-leaf subtrees connected to it. We will assume that no internal node in T and T' has degree two, and we will denote the maximal internal degree of all internal nodes in T and T' by id and id' respectively. Let v be an internal node in T, and let F1, ..., Fdv MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGgbGrdaWgaaWcbaGaemizaq2aaSbaaWqaaiabdAha2bqabaaaleqaaaaa@30EB@ be the subtrees connected to it, as shown in Fig. 6 We call these the subtrees of v. We say that v claims all butterfly quartets ab|cd where a,b ∈ Fi, c ∈ Fk and d ∈ Fm for i ≠ k ≠ m (see Fig. 7). With this definition, each butterfly quartet is claimed by exactly two internal nodes.