Correct answer: I < II < III
The nitrogen lone pair conjugates into the adjacent pi system. In (I) the lone pair is delocalized all the way onto the electron-withdrawing carbonyl, giving the C-N bond the most double-bond character and the shortest length. In (II) it delocalizes into an extended diene but with no strong acceptor, giving intermediate shortening. In (III) beyond the single C=C there is only a saturated chain, so the least delocalization and the longest C-N bond. Increasing bond length: I < II < III.