In order to determine how best to control the tree, we need to identify it, if possible. I think I've identified your problem as a Paper Mulberry (Broussonetia papyrifera). It's native to China and hardy in your region; the leaves in your picture are young leaves - you may see differently shaped leaves on the 12 footer next door. The attached link gives us one confirmatory point and one useful point.
This helps confirm the ID: "The tree can be weedy and fast-growing, spreading aggressively by means of root suckers, but it can be pruned when dormant (late fall or early winter) to control growth"
And this provides a potentially useful tip: "it has a shallow root system"
The tree is considered an invasive species across much of the South; not sure about Texas, and when I looked at Texas A&M's site it was terribly unhelpful. If it is invasive in Texas, the landlord may be legally required to remove it (laws on this differ by state).
So, how to control it? You have two options: physical barrier or chemical control. As noted in another answer's comments, chemically treating it may kill the mother tree, which doesn't sound like it'll be a bad thing (especially now that we know what the tree is).
I know that this works with shallow-rooted trees and shrubs because I've done it myself. It's a lot of work, but it's one-time work that you'll never have to repeat, and it's permanent.
- Find aluminum flashing at a hardware store. I recommend at least an 8" height. While you're there, buy a tin-snip if you don't have one.
- Dig a V-trench along the fence-line, with the straight edge of the trench just below the fence. You'll have to dig the trench the height of the flashing minus one inch. Trench the entire length of the fence all at the same time.
- Starting in a corner, lay out about 6-8 feet of flashing (more if you have a helper) and begin burying it, a foot or two at a time. Continue until you've buried all of the flashing.
My entire backyard is protected from my neighbors' weeds at this point; this method has saved me countless hours of work and frustration because one of my neighbors has a weed-pit for a backyard, with three different nasty stoloniferous weeds.
There is a possible negative here, specific to the mulberry. There is a good chance that the tree will sprout on the neighbor's side of the fence when its roots hit the flashing, so you could wind up with an unintentional hedge there. Since these are trees that can reach 50' in height, this could become a larger issue and could certainly damage the fence.
Ignore the posters who recommend RoundUp because it is ineffective on woody plants. The chemical to use is called Triclopyr and it's safer (for you) than RoundUp. It's usually sold as Stump and Brush Killer. Make sure you buy the concentrate, not the already-diluted product. The Ferti-lome brand that I linked to should have a brush inside the cap. It's easy to use - simply cut the stump or stumps to within 6" of the ground, then paint the top AND SIDES of the stumps with the chemical. The tree and all of its roots will die.
Easy, but not as much "fun" as burying the flashing. Not a permanent solution, though, as I expect the tree to return every year.
I forgot to add that you can cut the stump down to ground level a month or so after treating it, if you want to.