Ask Mr. Smarty Plants

Cordia boissieri (Mexican olive) is endemic to Texas and, as you can see from this USDA Plant Profile Map does not grow natively in Bexar County. We say this because of the very cold weather we have had in Central Texas the last week or so, and we wonder if you may be having worse problems than whether to prune,  but we hope your tree has managed to survive. It is supposed to be all right as far north as Austin but this has been most unusual weather. For this reason, we recommend you do nothing for a month or so. It is good to do any pruning of a woody plant (trees and shrubs) in the winter, but waiting until January and assuring the tree is alive might be a good idea.

The research we found indicated that it should be pruned only for shape, but did not say it disliked pruning. We also learned that some gardeners prefer to leave lower branches and allow it to be a bush. All indications are that if it loses some leaves because of frost that they will quickly reappear. In January, if you are unsure if it is alive, give it the thumbnail test. Starting at the highest branch you can reach, scratch off a small thin sliver of outer bark. If there is a thin layer of green beneath that, the whole tree is alive. If not, work your way down until you find some green. We suggest this test before you remove those lower branches just in case that is all that is alive and need to be left so the leaves can make nourishment for the tree and permit it to grow. If higher branches are dead, they will not regrow.

If you have established you have a living tree that does not require retention of lower branches, there should be no problem with cutting off those lower branches.