It is not a large tree so if it falls it probably will do little damage except to the fence. So there is little you can do but cut it back to the lot line and live with it. If it falls you may be able to sue for damages, BUT you will probably loose as you are responsible for what is or falls into you yard. (Most areas of US)

I would cut off flush to the fence and plant a tree of my own in front of the dead tree.

KR,did not know we are responsible for what falls in our yards…..maybe I shouldn’t have removed my huge Box Elder last fall…..it would’ve fallen into my neighbors yard…..☺️ But I am not an a-hole neighbor like Ontario’s. So I removed it. It was my tree after all.

Sherry, I absolutely agree with you 100%! A tree or a really fast and tall growing vine.

debra

Just for the record, I didn't call my neighbour any names.

About 2 months ago, a dead branch fell off the tree, I put the branch on his front porch with a note 'Time to remove the tree' but there was no response.

Keep the suggestions coming!

As for who is responsible for a down tree, and the damage it causes.

This is I believe comes from English Common law. BUT it can vary, and It may be different in other areas. To get the absolute facts, you should check with your local government authorities in your specific city, county, state, or country,

Let's focus on prevention, what can be done to prevent the issue from getting worse.

I am frankly surprised that after a decade of decline, there is no sign of disease or insect infestation that could spread to my neighbour's own trees/garden and to my and other neighbours' trees/gardens. An ugly tree with dead branches is not a strong argument to have the tree removed, especially if my neighbour does not see the tree!

I have several dead trees in the back of my yard away from everything. Why. Our yard is designed to attract birds and butterflies, and we do attract a lot of different birds, butterflies, and other wild life.

I think the ones we enjoy most is seeing the pileated Woodpeckers. The woodpeckers feed on insects often found in old dead trees. So to feed this type of bird we need dead trees. It is amazing to see this primitive looking bird tearing into a dead tree.