If the sub-tree is constructed from a binary rule rewrite X → Y Z, then the root nonterminal Y of some best sub-tree over some span (i, k) will generate break b k because Y is the highest nonterminal that covers word w k as the right-most terminal 4 .
the root nonterminal Z of some best subtree over some span (k+1, j) can not generate break b j because Z has a higher nonterminal X that also covers word w j as its right-most terminal.