Both species of privet are problem weeds. Chinese privet (Ligustrum sinense) tolerates quite deep shade and tree privet L. lucidum produces huge quantities of seed, dispersed by the birds.
Control: Drill a series of ten mm holes in the trunk and fill with glyphosate herbicide, ringbark and paint herbicide on the cuts or cut down and remove the whole tree. If the dead tree is to be left standing, remove the lower branches to allow access to the other weeds.