Table 2 Proof of four-point condition for double anchors. Four point condition for all 7 cases of adding {u,v} to a quartet tree, as shown in Fig. 4 (left side)
From: Anchoring quartet-based phylogenetic distances and applications to species tree reconstruction
| Â | \(L =D^{\prime }_{uv}[a,b]+D^{\prime }_{uv}[c,d]\) | \( R1 =D^{\prime }_{uv}[a,d] + D^{\prime }_{uv}[b,c]\) | \(R2 =D^{\prime }_{uv}[a,c] + D^{\prime }_{uv}[b,d]\) |
|---|---|---|---|
Case 1 | [β−f(e 1+e 3)]+[β−f(e 2+e 3)] | [β−f(e 3)]+[β−f(e 3)] | [β−f(e 3)]+[β−f(e 3)] |
Case 2 | [β−f(e 1+e 2+e 3)]+[β−f(e 3)] | [β−f(e 3)]+[β−f(e 1+e 3)] | [β−f(e 1+e 3)]+[β−f(e 3)] |
Case 3 | [β−f(e 1)]+[β−f(e 3)] | [β+α e 2]+[β+α e 2] | [β+α e 2]+[β+α e 2] |
Case 4 | [β−f(e 3)]+[β+α e 1] | [β+α(e 1+e 2)]+[β+α e 2] | [β+α e 2]+[β+α(e 1+e 2)] |
Case 5 | [β−f(e 2+e 3)]+[β+α e 1] | [β−f(e 2)]+[β+α e 1] | [β−f(e 2)]+[β+α e 1] |
Case 6 | [β−f(e 1)]+[β+α(e 2+e 3)] | [β+α e 3]+[β+α e 2] | [β+α e 2]+[β+α e 3] |
Case 7 | [β+α e 1]+[β+α e 3] | [β+α(e 1+e 2+e 3)]+[β+α e 2] | [β+α(e 1+e 2)]+[β+α(e 2+e 3)] |