yes bury wood, no absolutely do NOT use willow, unless you want willow growing there !!! period.
the smallest piece of willow can sprout and make a new tree
I have been burying wood as i have harvested in many of my beds..last year I dug up and moved a gob of jerusalem artichokes, and as I managed to clear each bed of sprouting babies..i finally dug out all the soil down about 2' and filled the hole with wood and chips and mulch and whatever I could find and then I put the soil back in..
unless you have permanent beds you can do this..even in permanent beds you can dig a hole and stick in a chunk of wood..
you can also use pieces of wood on top of the soil around plants to protect them and
feed them, I do that here..putting rotting logs around things like my baby nut and fruit
trees, etc.