Fig. 1

The three trees \(T_1\), \(T_2\), and \(T_3\) with common leaf set \(L=\{a,b,c,d,e\}\) have the (unique) common refinement T. Here, \(J(\rho )=\{1,2,3\}\) and thus, \({\bar{J}}(\rho ) = \emptyset \). The different symbols for vertices indicate which vertex u in the \(T_i\)s corresponds to which vertex u in T. Consider the vertex v highlighted as \(\blacksquare \). The corresponding vertices \(p_i(v)\) are shown in the respective trees \(T_i\). Here, \(p_2(v)=v\) while the vertices \(p_1(v)\) and \(p_3(v)\) in \(T_1\) and \(T_3\) correspond to \({\text {parent}}_T(v)\) and \(\rho \), respectively. Consequently, \(J(v) = \{2\}\) and \({\bar{J}}(v)=\{1,3\}\). We have \(p_2(v)=v\prec _T{\text {parent}}_T(v) = p_1(v)\prec _T p_3(b)\), according to Obs. 3. In this example, only the last case in Obs. 4 for v is satisfied, namely \({\text {parent}}_T(v)=p_1(v)\). Moreover, \(A(v) = \{v\} \cup \{{\text {parent}}_{T_2}(v)=\rho \} \cup \{p_1(v),p_3(v)\} = \{v,\rho , {\text {parent}}_T(v)\}\)