How things tie up N has been told,peated and repeated so many times I figure everyone already knows "how it is". For some strange reason wood and green plants where I live didn't get the memo. I've used and showed others how to use practices called huglekultur and keyhole garden where logs,limbs,twigs and leaves are piled similar to what you would to burn it then covered with soil and planted. Shallow rooted annuals do about the same first year as if planted in plowed ground. Second year all plants perform better and require less irrigation than those in plowed ground. Third year and following years it's off to the races,leaving plants conventionally planted in ground in a cloud of dust while using 75% to 60% less irrigation and fertilize. Sawdust is a different kettle of fish.
N robbing green plants? For as long as I can remember farmers around here have turned green cover crops under and made a living growing crops with less fertilize and irrigation than land minus cover crop. The only grass I bag while mowing my yard is from area near air conditioning condensers. I put the clippings in a bin with brown leaves from previous Fall and the mix gets too hot to hold a hand in.
BTW the resultant soil doesn't stick to tools near as bad when wet as native black clay.