Live oak trees are notorious for suckering from the roots. This is one reason why you see oak mots (clusters of trees) growing in natural areas. Suckering is increased by watering often around the tree or rototilling the soil beneath the canopy which disturbs the roots. There is little you can do for suckering except to mow them of on a regular basis and to not disturb the soil or water any more than you have to. Landscape fabrics are partially effective but need a heavy mulch over them or the shoot will lift the fabric up as it grows.
Spanish moss is not a problem for the tree but if you the look of it you can use a long pole with a wire hook on the end to pull it out of the tree and thus reduce the amount of Spanish moss considerably.
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