It must be remembered that the bark contains the phloem in its innermost region. The circumcision of even a thin layer of bark completely disrupts the movement of organic nutrients from the leaves and upper stem to the lower stem and roots. In a clear sense, the roots and lower stem would be starved and their functions suspended. Survival would be impossible.
On the other hand, a deep cut along one side of the trunk would disrupt completely both nutrient flow in the phloem and fluid movement in the xylem, but only in the narrow band of elements actually destroyed. The rest of the trunk would continue to provide a conduit for necessary fluids and the sugars and other organic foodstuffs manufactured by the leaves. Sometimes, it is possible for trees to survive even though their heartwood has rotted or burned. Although this diminishes their strength, it does not reduce their ability to transport nutrients, and so they survive.