Recently, some research also showed that treating the parse tree as latent variables (Loehlin, 1998) can benefit the BTG tree inference but for preordering (see Figure 2).
the BtG terminal rule (t : X → f /e) is used to translate the source phrase f into the target phrase e while the straight and inverted rules (S : X → [X 1 X 2 ] and I : X →< X 1 X 2 >) are used to concatenate two neighbouring phrases with a straight or inverted order as following: where • stands for concatenation between strings.