Plant or tree suckers are an effort by the plant to grow more branches. It is preferable to prevent tree suckers than to remove them.
You can control tree suckers by keeping the plant in good health; avoid additional stress on the plant caused by drought, overwatering, diseases, or pests.
Regularly prune the bottlebrush, but avoid over-pruning as it can stimulate tree suckers.
If you notice any tree suckers, remove them immediately as they will divert the energy from healthier branches. Use a clean pair of pruning shears and cut the sucker close to the tree. Leave the collar to help with recovery.