There is some sort of vine that seems to wrap around small trees and over time pull them to the ground, killing them in the process. Is there anything I can do to rid the vines and save the trees?
You seem to have two non-native Invasive vines here. Both need to be removed.
The first photo is Oriental Bittersweet, a non-native Invasive plant from Asia. It is one of the worst invasive vines in the U.S. and Maryland. It gets orange berries in the fall which birds spread, so you'll often see seedlings coming up under trees because the birds drop the seeds when they perch. The leaves are an oval.
The roots are orange. This makes them easy to identify when pulled. They pull easily when young. When they get too big to pull, cut them down and immediately treat the fresh cut vine stump with an herbicide containing glyphosate (in Round-Up and many other products) or triclopyr. You have to kill the root, so it it resprouts, pull or retreat. (Glyphosate kills everything, so don't get it on other good plants or tree trunks.) You don't have to pull down the vine still hanging on the tree after you sever it. It will slowly decompose.
Here is a link about Oriental bittersweet from the Weeds Gone Wild website, a great source for how to handle all the worst invasive plants:https://www.nps.gov/plants/ALIEn/fact/ceor1.htm
The second photo is Japanese honeysuckle, another noxious non-native Invasive vine. It, too, smothers and pulls down trees as Oriental Bittersweet does (though not as heavy as bittersweet). Kill it the same way you would Oriental Bittersweet. There is also a fact sheet on Japanese honeysuckle in Weeds Gone Wild. Just click on Vines. https://www.nps.gov/plants/ALIEn/factmain.htm
It is not terribly hard to get rid of these vines, just stick with it. In the future, walk around and look for the seedlings of these two vines and pull them when young, before they get a chance to damage your plants.