You didn’t mention what kind of volunteer tress you have, but I think they will detract from the overall appearance of an old live oak tree Quercus virginiana (Coastal live oak), and should probably come down.
At this time, they are competing with the oak tree for water and nutrients. As they grow larger and penetrate the canopy, they will also be competing for sunlight. Neither of these situations benefits the oak tree.
Another thing that happens under oak trees, and you may be experiencing this, is the production of numerous root sprouts. Check out this link to learn more about these pesky little plants.