It isn't necessarily required to plant trees back. It depends on the type of management and cutting being done.
One strategy is to remove the highest value trees from time to time, as they tend to be larger and removing them creates space for smaller trees to grow larger. Such harvests will also remove smaller trees that are unlikely to increase in value (often sold as paper pulp or wood chips, not necessarily discarded). This creates space for new trees to grow.
At the extreme you do have companies cutting down everything and using herbicide cocktails to prepare the ground for planting.
One strategy is to remove the highest value trees from time to time, as they tend to be larger and removing them creates space for smaller trees to grow larger. Such harvests will also remove smaller trees that are unlikely to increase in value (often sold as paper pulp or wood chips, not necessarily discarded). This creates space for new trees to grow.
At the extreme you do have companies cutting down everything and using herbicide cocktails to prepare the ground for planting.