We have come across a property where the boundary is up close to the house. Next door have trees and shrubs that have taken over and grown up and onto the roof of the house.
It made access impossible to the gutter and we could not complete our gutter clean.
Having trees and debris in and on your gutter can cause multiple issues. The weight can cause the gutter to sag or even come away from the house. Constant moisture would be present, together with the corrosive sludge, would contribute to the premature rusting of the gutter.
Talking to your neighbour is your first course of action.
A discussion of what can be arranged that is accommodating to both parties for the removal of the tree limbs/brush, where there is considerable expense, will aid in keeping an amicable relationship between neighbours.
Yes you are entitled to trim back your neighbours tree and throw the rubbish on their side of the fence.
Before doing this you should check what rules apply at your local council.
Do you want to live happily with everyone?
This was a big job to remove to foliage but it is going to help preserve the life of this gutter.