Nitrogen depletion only becomes a problem when fresh green woody material is mixed into the soil. Spreading it on top of the soil will require more time for it to decay, thus less nitrogen from the soil is consumed. Probably any that is consumed near the surface, would be replenished by normal rainfall, assuming there is such a thing as normal rainfall in your area! Green leafy vegetation in the mix will release Nitrogen as it decays, so if there is enough of it, it will probably offset the small amount consumed by the wood chips.
Rb

If you want to encourage some small animals to live in your woods, you can leave some of the brush in piles. Small critters like to live in, under, and around such things. Just an idea.

FataMorgana

There's a good amount of green mixed in, so it should be alright.

FataMorgana, there is forest all around my property so the critters have plenty of hiding places.

The Chinese privet got so thick and I got tired of not being able to walk through my own land, I had to do something. As I'm clearing out the privet, I'm planting new native trees.

More power to you on the battle with invasives. I spent several hours on Sunday using a "Weed Wrench" prying invasive privet shrubs out of the ground (the cut ones just kept growing back).
Don't worry too much about the nitrogen depletion since your are not tilling in the chips. Also if the material you cut was live it might have a bit more nitrogen than expected. Another possibility is to have a designated wood chip pile where some of the initial breakdown can happen for a year or two and then distribute the chips - this is what I do with the large loads of chips that the arborist leaves for us after doing work. You can also monitor the pile for any sprouting of invasive seeds that got in there. You might have also noticed that piles of fresh woodchips can heat up on their own and get the decomp started off nicely. You can use the wood chips later to mulch the trees you are planting. I think there is some justice in that.

Yes, I have a big pile of chips leftover from last year, and I did notice plenty of heat coming off of them after I piled them up. I turned them a few weeks ago and the chips have turned completely black. I also noticed lots of earthworms in the pile, some of them were the size of small snakes. It's just so much easier to spread them where I'm working, rather than trying to pile them up though.

The only way I stopped the privet from coming back was to squirt some full strength Brush-B-Gone on the stumps immediately after cutting. My biggest battles are Chinese Privet and Chinese Tallow.

O.K. Just distribute the chips as they are being made. Only a portion of the surface area of each chip is in contact with the ground and potentially tying up nitrogen. The green parts of the chips will give some nitrogen anyway. I wouldn't worry about it. Battling invasives requires compromises you would not make in an estate garden.
I have almost every invasive except Kudzu and that is already just a few counties south of me (NYC area - really!).

Search "ramial chipped wood" on your search engine.

Here is a link that might be useful: Ramial Chipped Wood

Thanks, there's lots of info in that link.

I don't think a small reduction in the amount of available nitrogen in a woodland is a bad thing. Invasive plants tend to thrive in nutrient rich, disturbed areas. Adding a mulch to stabilize soils and reduce the available nitrogen might help slow the spread of invasive plants and help you get an upper hand. I have heard of people who are restoring natural environments using "reverse fertilization" by spreading sawdust on the soil to reduce the level of available nutrients and thus limit the spread of invasive plants. Most of our native plants are adapted to low levels of nutrients. The same is true of most native plants in other parts of the world - it is the few ruderal plant species, adapted to quick growth in disturbed, nutrient-rich soils that tend to become invasive species when they get to areas to which they are not native.

Also, I'll add that I think brush piles are a great addition to any woodland, regardless of the surroundings. I have several small patches of woodland amid brushy fields, a powerline right of way that is thick with brush, and some weedy fields that are quite dense. Even with all of the brush around here, brush piles are heavily used and seem to attract more birds to my yard than I would have otherwise. They are particularly well used by birds that visit my feeders in the winter.

By the way, if you let your cut brush dry out for a few days or weeks it will be easier to chip.

I do have plenty of Wax Myrtle growing here, which produces nitrogen through it's roots (similar to autumn olive and legumes). That may actually counter any losses due to the chips. I just thought about it.

I have been using wood chips in and on the soil for years and have never noticed a nitrogen deficiency yet. Whenever I till I usually add an inch or two of chips, then plant, then mulch with chips. The soil has gotten significantly better every year. Yes, the chips absorb nitrogen, but sooner or later it gives it back. I haven't seen any yellowing of plants in all the years I have been doing this. Whoops! One exception. I had a truck load of fresh sawdust delivered which I promptly tilled in and planted right away. Those plants were a bit yellow for awhile then turned a nice green. The sawdust was fresh and had a lot more surface area to absorb the available nitrogen quicker than chips.

Here's a good parallel. ;-) I was in a bar in Montana once that had salt shakers on the bar for people who wanted salt in their beer. Unlike normal salt shakers, these contained rock salt and had larger holes in the lid. I asked why they didn't use regular salt.
"Oh, that's flash powder. You put that in your beer and 'woof', you have flat, salty beer. With Rock salt you have a little salt taste, but the beer doesn't go flat."

Last year, I read a current article for arborists, and it mentioned that top application of chips does not result in "nitrogen robbing". Since that's a mulch layer, not an ammendment.

Years back, I had other pruning / tree services dump about 50 large truckloads of chips on my mom's 2 acres - worked great.

Now that this reminds me, I'll post in in the soils section of the link below in the next day or two...

Here is a link that might be useful: Turf to Trees