you can buy a whole roll of plastic coated tie wire at most hardware stores, you can cut it to any length you may need. Thats what I use and it works great. I also recommend a pothos and maybe a schleffera for live plants.
if you want to run your branches horizontally, you can use pushpins to keep the branches up. Just attach them on the outside of the screen and into the ends of the branches. Works great and keeps the branches secure and from falling!