Hi Ed, Kimberly here from landscape solutions for you. Thank you for asking the question on your Weeping Yaupon Tree.
The general rule of thumb is to prune the entire tree 20 percent. This can be done twice a year and in the first of pruning, three times. If you prune from the bottom, you are going to promote growth. You will want to prune from the top, always cutting the branches at an angle. Keep the tree in its natural form by pruning it from top to bottom.
You live in a warm climate, so I would recommend waiting until mid to late August to do your first prune. (or even the first part of September) You can go ahead and prune again next February, and then probably in June again. You want to take off about ten feet, and it is best to do this gradually, that is why 20 percent is recommended. Make sure you do water after pruning, even with an established tree.
I hope this helps some and let us know what you decide and how you make out. You can trim some of the lower branches near the ground off, but it will promote growth. If you need more information on this or any other garden or landscape topic, feel free to write about it here. Have a great day!
Kimberly