Non-toxic barrier for invasive tree for raised vegetable beds

Hi,

I'm looking for suggestions for a non-toxic material that I can put under deep raised vegetable garden beds that an invasive tree won't be able to come up through.  For years my yard been getting taken over by the tree of heaven.  It's invaded the area where some of my raised beds are.  I'm look for something I can put on the ground, and then build deep raised beds, 18 - 24" deep, on top of.  

Any suggestions would be appreciated.  

Info on the tree from hell if you're not familiar with it:

https://www.fs.usda.gov/Internet/FSE_DOCUMENTS/stelprdb5410131.pdf

Thanks

How long do you need the barrier to last?  Would it need to allow water to drain out?

I'm imagining a sheet of 3/4" plywood (not treated) would eventually rot but would hold out roots for a decade or so.  4 layers would last longer.  But it would hinder drainage.  

Alternately you could try a series of layers of cedar boards.  One layer running E/W and touching as tightly as possible.  Another layer on top running N/S and screwed to the layer below.  Another E/W layer and one last N/S layer on top of it.  That much cedar would last for quite a while and the cracks may allow enough drainage without hopefully letting roots invade.  Tongue and groove cedar would probably seal out roots even better but drain less.

Or a sheet of bare steel sheet metal.  It would probably remain solid for a decade or so.  The thicker it is the longer it would last.

Air.
If you are creating a barrier between the bed and the rest of the earth, you might as well put it on a platform.
Cement blocks layed on their side's.
I would use sheets of masonite you build it on the ground.