cutting off some branches on your side won't help with the leaves that will blow over to your side from the rest of the tree!!.... and it would make matters stinky if the neighbor didn't like your trim job.... why not leave the tree alone and come up with something intelligent to keep the leaves out of your pool..... btw... it's almost September.. fall... leaves drop, ya know?.... I have to put a stretch of bird netting over my pond to keep falling leaves out... perhaps you could come up with something like that for the pool?.... or kinda like those nets behind some sports?... they're loose and floppy, not like batting cages, but still stop wayward balls?... something like that would keep a lot of leaves from going in the pool...???....
the both of you should be returning those leaves to the soil with all their nutrients ... make compost piles!!!.... don't bag them and let someone take them away!!!.. and then turn around and put fertilizer on everything!!.... make compost with them!.... if both of you have piles, those leaves will be a blessing instead of a pain in the tookus!.....and... with a mulching mower, you can get some of them back into the lawn for good stuff there, too!.... get a lawn vac if it's too much to rake... you can rent them, ya know?..... shoot... I gotta go steal the bagged leaves from everyone else's curbs cuz all I have is pine trees and I need the leaves for MY compost!!!... quitcherbiotchin' and USE "EM!!!..