Looks like it's my turn soon ! It's brewing but nothing has really happened yet.
We moved here in 2006. Neighbour in question has been here far far longer, they *possibly* moved in when the houses were built, in the late sixties. Married couple I guess in their seventies now. The water table is very high here. A couple of people have said the road follows the course of an old stream but I've never investigated it. We do have a massive willow tree in the front garden and an undated old photo of the house shows two which does back up the stream thing.
Our gardens would turn to a marsh whenever we had heavy rain. The neighbour in question, their house is at the lowest point of the road, and their back garden would literally be under water every time.
In 2015 we had our front & rear gardens re-done, including installing a lot of extra drainage. The back garden has a load of perforated pipe installed feeding into the drain which heads off away from the back of the house to who-knows-where. The front garden has a load of soakaway crates feeding into the drain that heads out into the road. This made a huge difference, our garden is no longer a marsh and next door only under water after the heaviest rain.
Roll forwards a few years.
Neighbour, let's call him "Toni", became obsessed with my cars. Seemingly if one of them is idling on the driveway for more than 30 seconds he would appear to put something in the bin, or do some pruning, or something equally inconsequential, and scowl at me. Part of the problem here is that his kitchen has a door straight into the back of his garage and he likes to leave the garage roller door and the kitchen door open a lot of the time. So any noises or smells from outside reach his kitchen quite easily.
This culminated one evening last December when I was having enormous trouble getting the Galaxie to run well enough to drive it onto the trailer. I'd sold it and needed to deliver it to the buyer the following day. It had never acted up like this before - sod's law. I'd had it running a while to try and get it properly warmed through, when he came out shouting at me, claiming I do this every couple of days and he can taste the fumes in his house, blah, blah. I was pretty stressed out with the car, it was dark and cold and I was NOT in the mood to discuss it at that point so I stressed that it's not every couple of days, I need to get the car on the trailer, and if he wants that conversation it needs to happen another time. I've not spoken to "Toni" since then but I've been mindful of keeping my cars away from that side of the drive. My wife always chats to him & his wife if she sees them though.
We had a lot of rain a couple of weeks ago and "Toni's" garage flooded. He had Dyno Rod out to clear the drains.
"Toni" beckoned my wife this afternoon when she got home and he says that:
* Our willow tree has broken his drain (that goes in pretty much in a straight line from his kitchen, under his garage & driveway, to the road) and the roots have grown into the pipe, blocked it, and caused the flooding, so I need to fix it.
* He wants us & the neighbour on the other side of his property to pay for upgrading the drainage in the back garden, as according to him it's runoff from our properties that is the problem.
My wife is very nice and would happily say yes to all this for fear of offending. He's not tried asking me yet

So getting to the specific points here:
* Are the roots of my tree outside my property my problem ? I know that overhanging branches are not strictly my problem but he has moaned about those enough times in the past and I've always trimmed them for him as it was only a ten or fifteen minute job.
* It's not runoff, it's not like there's a waterfall cascading into his garden, it's just groundwater rising after heavy rain, and being the lowest point his property is the most affected.
My feeling is that I can legitimately tell him to go do one. What does everyone else think ?
(to be clear - his name is not "Toni", it's "Tony")