When we moved in my husband and I introduced ourselves to our backyard neighbors because we were clearing out our green space of invasives and replacing them with natives. This particular neighbor very dramatically "surveyed" the area with a buddy of his (while my husband and I were working at pulling out poison ivy in the area) and not so nicely explained not to touch his side. That his how I know for a fact the tree belongs to him.

Now I am faced with the dilemma of not being able to let my kids play in our backyard because of a tree that could fall at any moment. I understand its winter and might be hard to do something about it now but it has been dangling since the fall and he has not once mentioned it to us. My kids miss playing in the yard and part of why we moved here is because of the fact that there are woods for my sons to play in. Now they are playing in the street.

I'm afraid to ask the neighbor of his plans without causing an argument. What would you do?

If he objects to your crossing the boundary to take down one of his trees, tell him that in that case, he will have to remove it at his own expense. The monetary alert might persuade him to let you do it and pay for it yourself.

At least, it's worth trying.

I have a similar problem with a neighbor who doesn't care about anyone else and doesn't respect my property. He is more blatant about it though, and has actually had a large drain pipe spilling over the wall into my property, had been throwing yard waste and other debris into my garden, and most recently placed cinder blocks, boards, a trash can and a ceiling fan on my sidewalk! I have had to confront him on all these things, and although I was polite, I was also very direct in telling him what I thought. Those issues have been corrected except for the refuse on the sidewalk, since I only told him about it yesterday. That's because I didn't see it there until the snow melted. We'll never be friendly, but I figure if he's going to be a jerk, then I have no reason not to let him know how I feel when he does these things. i.e. nothing to lose. It's not like you'd make an enemy or something if he's already acting like one!

So I say let him know of your concerns, in a definite but friendly manner. If he removes the broken limbs, great, and if not, he probably wasn't going to anyway.
Good luck!

But honestly I would be more inclined to just wait for nature to take its course and steer clear of the danger zone in the meantime, especially keeping your kids from playing under the leaning tree branch. Surely they can play in the rest of your yard rather than in the street? Automobiles might be a greater danger than the offending tree, depending on the traffic situation.

Also if the woods in question belong to your neighbors, they may be distinctly unhappy about having your children playing there. I would not welcome children playing in my woods or on my land unless their parents were visiting at the time, in part because of the liability, and also because of the invasion of privacy. But I also would object to adults digging, or requesting to dig, plants on my land, invasive or not.

Good luck. Maybe your neighbors will surprise you by being accommodating about your removal of their leaning branch.