Neighbour complaining about tree

Update. Forgive me everyone I don't know if I'm only supposed to reply to people individually. I'm very grateful to everyone's input.
Tree now photographed. It's approximately first floor height so that's
approx 15ft. On the surface healthy. Coming off Main trunk branches are heigher
than height of fence (5ft) so any of those favouring neighbour's garden are high up. a 'knuckle' at about 10ft height off main trunk appears to have been cut off but needs further verification. Basic description Main trunk splits into three vertical thick branches with the
obvious 'fanning' smaller branches coming off them. Two of these thicker branches (making a V) are definitely well within my Mothers boundary, the third more the neighbour's. Cutting that thick branch would more than likely make tree uneven as whatever leafing ensues over Spring and Summer with small branches will only be on the two 'V' branches on my Mothers side.
Tree is Sycamore and handsome, quite majestic in the skyline despite suburban cut everything tall around in other gardens look! Entire neighbourhood streets have many at the front of houses, so obviously the original design when houses built roads set out.
Person taking photos noticed new decking next door. Without the third
thick branch 'their side' (assuming it would be safe to cut it off which is a bit like cutting tree in half vertically) the leaves from tree will still blow in
neighbour's garden. Could be this is their issue.
Parents owned house almost 40 yrs, tree was there then but skinny.
Neighbour's there 6 yrs approx.
I'll look on link suggestions, from the outset it hasn't made sense to me
that anyone could cause something not dangerous to become so.
I don't think right of light is an issue if I've understood correctly as
No windows are covered and trees been there a very long time.
Again thank you all so much.

Davesnave wrote: »

If it's only around 15' high after 40 years, it's not a sycamore, though it might be a maple, which is in the same family.

Fifteen feet is no height for an ordinary non-grafted tree; many of the native trees we planted in the winter of 2014 have exceeded that already. Whatever it is, it's probably suited to a residential environment and unlikely to put on rapid growth now.

To be honest, I was imagining a tree of 30' plus, which would be of greater concern if lop-sided. We have about 8 or 9 oaks here of the 15' variety, preserved when we laid a roadside hedge, and most of those are uneven, because passing lorries and double decker buses limit growth on the road side. We have no worries about them falling at the small size they are now.

Whatever the neighbour does, if they stay within the law, your folks are unlikely to end up with a dangerous tree IMO.

Thanks yet again for your input. I may be slightly out with the 15ft height I just calculated approx based on the top being level with first floor bedroom window almost from the photographs, either way it doesn't exceed 20ft and most definitely isn't 30ft.
This is such a learning curve, I've found out 'right of light' which I kept reading about on various sites would only apply
if tree differently positioned, was over 30ft and Evergreen so no
ticks in those boxes.