Table 3 Cases for an associated triplet \(L_{t} = \{t_1, t_2, t_3\}\) of a shared solo quartet in recursive equation (2). In each case, the quartet is counted exactly once, as shown in column \(\sum\)
From: Median quartet tree search algorithms using optimal subtree prune and regraft
Case | \(S_4({B} \overset{u^{\uparrow }}{\circ }P, R)\) | \(- \varphi _{u^{\uparrow }}^i\) | \(- \varphi _{u^{\uparrow }}^r\) | \(\varphi _u^o\) | \(\varphi _u^r\) | \(\sum\) |
|---|---|---|---|---|---|---|
\(L_{t} \subset L_{{B}^\vee _{u}}\) or \(L_{t} \subset {L_{{B}^\vee _{u^{\uparrow }}} \setminus L_{{B}^\vee _{u}}}\) or \(L_t \subset L_{{B}^\wedge _{u^{\uparrow }}}\) | 1 | 0 | 0 | 0 | 0 | 1 |
\(L_t \not \subset L_{{B}^\vee _{u}}\), \(L_t \not \subset {L_{{B}^\vee _{u^{\uparrow }}} \setminus L_{{B}^\vee _{u}}}\), and \(L_t \subset L_{{B}^\vee _{u^{\uparrow }}}\) | 1 | \(-1\) | 0 | 0 | 1 | 1 |
\(L_t \not \subset L_{{B}^\vee _{u}}\), \(L_t \not \subset L_{{B}^\wedge _{u^{\uparrow }}}\), and \(L_t \subset L_{{B}^\vee _{u}} \cup L_{{B}^\wedge _{u^{\uparrow }}}\) | 1 | 0 | \(-1\) | 0 | 1 | 1 |
\(L_t \not \subset {L_{{B}^\vee _{u^{\uparrow }}} \setminus L_{{B}^\vee _{u}}}\), \(L_t \not \subset L_{{B}^\wedge _{u^{\uparrow }}}\), and \(L_t \subset L_{{B}^\wedge _{u}}\) | 1 | 0 | \(-1\) | 1 | 1 | 1 |
\(t_1 \in L_{{B}^\vee _{u}}\), \(t_2 \in {L_{{B}^\vee _{u^{\uparrow }}} \setminus L_{{B}^\vee _{u}}}\), and \(t_3 \in L_{{B}^\wedge _{u^{\uparrow }}}\) | 0 | 0 | 0 | 0 | 1 | 1 |