Complete girdling ultimately kills the tree due to

Complete girdling ultimately kills the tree due to Starvation of roots.
Girdling also called ring-barking, is the complete removal of the bark from around the entire circumference of either a branch or trunk of a woody plant. Girdling results in the death of the area above the girdle over time.
Girdling results in the removal of the phloem, and death occurs from the inability of the leaves to transport sugars to the roots. In this process, the xylem is left untouched, and the tree can usually still temporarily transport water and minerals from the roots to the leaves.