Dear Russel,

Thank you for your question! Vines can be a tricky proposition, especially if they are already going on something as hardy as a walnut tree.

In my experience, cutting an inch sized gap into the vines near the base will cause the portions that are higher up to whither. It's hardly a permanent solution but it will keep them from choking out your trees as they grow themselves.

For more permanent steps, this article offers some solutions:

http://homeguides.sfgate.com/kill-ground-vines-harming-trees-26599.html

Best,
Daniel Hubbell
if you have any other questions feel free to email us at info@mylandplan.org