Hi chapais! Welcome to NGA.
I'm very, very "down" on adding salt, even unintentionally via "salty" manure. So see if others speak out against it before taking my advice to heart.
I would never intentionally add salt anywhere near a garden, or even a lawn, or anywhere upstream from anywhere that anyone would ever want to grow plants.
Salt is very soluble, so first it would wash out of your trench, then kill the surrounding lawn, perhaps even damage the tree, then damage whatever soil is downstream (based on the flow direction of ground water.)
It would eventually flush out of your neighborhood and go "salt up" someone else's yard.
There are entire regions where agriculture is more difficult or no longer practical because of salinized soil. Maybe it is only a BIG problem where there isn't enough rainfall to flush all the salts OUT and into a river or ocean.
1.
I would suggest some thick plastic, like 4 mil plastic, to line the trench, but I think I read that bamboo roots can pierce that. Maybe trees can, too.
2.
>> Big roots find their way around the outside perimeter of the beds not far below the surface. Smaller roots then find their way in by forcing themselves through the plank joints.
Maybe buying a tube of construction adhesive and using it like grout between the planks would keep them out?
3.
It is possible (but I'm not sure this works with trees) to make a trench unattractive to roots by filling it with coarse gravel. Water drains out instantly, leaving no water behind for roots to follow. In containers, this would be like "air-pruning plant roots". But does that work with tree roots in gravel? I don't know.
4.
I've read that some roots avoid copper, so painting the outsides of the walls with some paint supplemented with copper might work. Or copper foil glued over the joints.
5.
The "guaranteed" solution is one proposed by some bamboo fancier who had the gall to call this "easy": "just" dig a moat and fill the moat with concrete. In your case, I guess the "moat" would have to extend under the entire bed or else go so deep that tree roots would never turn up after diving under it.
Since unused fertilizer and organics will be leaching down out of your bed, I would expect tree roots to do somersaults in order to follow them back to their source and steal all of them!
6.
There ought to be an intermediate solution, like deepening your trench a little and then lining the side of the trench that is up against the bed with something solid enough that tree roots can't penetrate it, but affordable. ("Affordable" might mean "see what's available at a Habitat for Humanity Restore", or scrounge-able elsewhere.) Very heavy plastic? Corrugated fiberglas? Phoney wood?
7.
If you don't care about the tree at all, you MIGHT try relying on an annual "haircut" on the tree roots. Keep your existing trench filled with something loose like gravel, but once per year, dig down deeper around the moat and CUT any tree roots that have crossed your Line Of Death.
Or cut a second trench between the bed and the tree, farther away from the bed, so that you might only need to cut the tree roots deeply every other year.
That will annoy the tree every year and might be bad or very bad for the tree, but how many feet can a tree root grow in just a few months?
(I wonder whether it would work better cutting roots in spring, or in fall?
I'm just guessing at solutions, so don't hesitate to say "THAT won't work!"
I tried Googling "tree root barrier" and it was suggesting multiple alternatives even before I got "barrier" fully typed. There are so many different products available that my guess is that NONE of them are both effective and affordable. If any one of them were both cheap and effective, there wouldn't be so many options about "depth" and "cost" and "systems". And there would be no "patented mechanical guides" or "A Review of Root Barrier Research". And there wouldn't be a Google suggested search phrase "do root barriers work?"
But looking at the products offered may give you ideas suited to your situation, or reasons why some approaches are seldom used.
I hope someone suggests more attractive solutions!
Eventually someone is bound to suggest the method that DOES work: "move the bed away from the tree at all costs", but I figure that, if that were an attractive option, you would already have done it to get away from both roots and shade.
When tree roots invaded my compost heap, I moved the heap. Or, rather, I abandoned the old heap since it would have been easier to get a bone away from a pit bull than to recover any soil from those tree roots.
