To me, nitrogen-fixing is a complicated subject.
The best way to use the nitrogen-fixing from those trees is to use the trimming as green manure.
I have read that using this a a green manure is the best way to get the nitrogen-fixing.
I hope these threads will help you or others:
Bryant said, N fixers generally utilize a bacteria that forms nodules on the roots for their home and these bacteria gather the nitrogen that is pumped down to them from the leaves.
For most plants it is the leaves that do the nitrogen gathering from the air that passes through their stoma as they go through the process of respiration during daylight hours.
https://permies.com/t/38055/legumes-release-nitrogen-continuously-killed#922542
Daron said, If you plant perennial nitrogen fixing plants my understanding is that you won't get much benefit during the first year. But leaf litter, any chop and drop material, etc. will steadily build soil and increase the nitrogen level.
https://permies.com/t/77546/nitrogen-fixing-plants-work#638360