A graft that does not take, or a branch that is broken or cut off the vine, dies. The branches must be attached to the vine to bear fruit. A branch that is isolated from the vine cannot live. 

As we consider the metaphor of pruning, it is about death AND fruitfulness.  Any branch that is cut off will dry up. There will be no life flowing through it. If, however, the Gardener has left a bud or two, the tiny buds will burst forth with new life, carrying the DNA of the vine to produce fruit.