From Longman Dictionary of Contemporary Englishdie back phrasal verbHBPif a plant dies back, it dies above the ground but remains alive at its roots → die→ See Verb table
Examples from the Corpus
die back• But a fortnight on, it had turned brown and almost died back.• Later, gum will ooze from the bark and branches will die back.• Cut back any plants around pond edges that are dying back, and compost them before they pollute the water.• During its resting period in Winter, i.e. October-February, the leaves die back, and during Spring sprouting takes place.• The frost causes the bracken to die back and thereby opens up the canopy to let in more light.• Banana plants will die back in a freeze but usually return in the spring from the rootstocks. 