Imagine we have a tree growing and then we cut-off a branch from it. Do the cells in that branch keep on growing/multiplying until the stored resources (in the branch) get depleted completely?? If cells don't grow, then what instant action happens which prevents the cells to grow further? What prevents it?
Depends upon the tree and the conditions.
Pussie willow shoots (especially green ones) can be placed in a container of water and will actually grow roots. These can be planted in the ground to form new trees or bushes.
Weeping willows do the same.
Experiments have been done with whole evergreens to determine the amount of transpiration (of water vapor) from their needles (the trunk of the tree being placed in an aqueous solution of dilute minerals).
Based upon the oxygen gas generated if the tree is in sunlight), the tree does continue to live for some time.