Fig. 3

An illustration of the derivation \(\mu\) from the PQ-tree T to the substring \(S'\) of S, where \(S'=S[3:8]\), under the one-to-one mapping \({\mathcal {M}}{}\) (\(\mu .o\)) with \(\mu .del_T=2\) deletions from the tree and \(\mu .del_S=1\) deletions from the string. The start point of the derivation (\(\mu .s\)) is 3. The end point of the derivation (\(\mu .e\)) is 8. Notice that \(S_{\mathcal {M}}{}=F(T')\) and \(T \succeq _2 T'\), which means that \(S_{\mathcal {M}}{}\in C_2(T)\). A The derivation \(\mu\) applied to T resulting in \(T'\): reorder the children of \(x_4\), delete leaves according to \({\mathcal {M}}{}\) (delete \(x_5\) and \(x_6\)) and perform smoothing (delete \(x_7\), the parent node of \(x_5\) and \(x_6\)). The root of T (\(x_{11}\)) is the node that \(\mu\) derives, denoted \(\mu .v\). Also, we say that \(\mu\) is a derivation of \(x_{11}\). The nodes \(x_5\), \(x_6\) and \(x_7\) are deleted under \(\mu\). The leaves \(x_1,x_2,x_3,x_8\) and \(x_9\) are mapped under \(\mu\). The nodes \(x_4, x_{10}\) and \(x_{11}\) are kept under \(\mu\). B The derivation \(\mu\) applied to \(S'\) resulting in \(S_{\mathcal {M}}{}\): apply substitutions and deletions according to \({\mathcal {M}}{}\). The substring \(S'=S[3:8]\) is the string that \(\mu\) derives. The character \(S[4]=S'[2]\) is deleted under \(\mu\). The characters S[3], S[5], S[6], S[7] and S[8] are mapped under \(\mu\)