Ask Mr. Smarty Plants

USDA distribution maps show two species of hackberry occurring in Travis County; Celtis laevigata (Sugar hackberry) and Celtis occidentalis (Common hackberry). Chances are that you have the sugar hackberry. Comparing the leave morphology from this link for the sugar hackberry with one for the common hackberry can help you determine which one you have. However, knowing the name of the plant is not an essential part of solving the problem.

The saplings that you are dealing with could either be seedlings or root suckers. The trees produce a prodigious number of drupe-like berries that are favored by many birds. The birds can digest the fleshy part of of the fruit but not the hard seed. These they excrete and help propagate the plant all over the place. As long as the tree is alive, you will probaly have seedlings.

The suckers grow directly from the root, sometimes in response to stresses such as drought, over-watering,  disease attack or woeful neglect. They are a means for the roots to increase the amount of photosynthesis that takes place and thus the amount of food that they receive. If you want to keep the hackberry tree, using herbicides on the suckers can eventually kill the tree as well as endanger your roses.

The most effective yet labor intensive means of controling your saplings is to continue to cut them as they appear. If they are seedlings, the repeated cutting will eventually cause the roots to die. If they are suckers, check on the health and care of the parent tree in order to reduce any stress that might be occuring.

These links from eHow and gardeningknowhow  have information about suckers and suggestions for controlling them.

Another source of help is the Travis County office of the Texas AgriLife Extension Service.