The parts left in the trees will die as long as no piece left connecting them to the soil. The exception might be if the vines have managed to actually root up in the canopy in pockets of debris. For English Ivy it is a good idea to cut the stems twice a few inches apart so there is a big gap which the vine cannot regrow across. It might be sensible to do the same with trumpet vine (I am assuming this is Campsis radicans.) But the cut off base of the plants will regrow and it will continue to put up suckers.

Thank you very much. This answers my question completely.