We put up a chain link fence in our back yard about 10 years ago, with the permission of the neighbors because we have a dog. I have always mowed and trimmed both sides of the fence to keep it looking nice. Now one of our neighbors is pulling up volunteer trees from our commons area and planting them inches from our fence, some of the branches are even poking through the fence and the trees are only about 2 feet tall at this point, the last tree will block all access to our back yards if left to grow. I can no longer maintain the fence line and our fence will be damaged within a few years, what are my rights?

Is the fence on your property... and if so how far from the actual property line is it. At the trees planted on the other side of the property line?

If the fence is in fact on your property line... you are in your rights to trim branches back to that property line assuming you don't kill the tree doing so.

If the trees were planted on your property they are your trees to do what you want with.

If all of this is actually on common property... you really can't do anything on your own. I suspect this might be the case because you don't need permission of neighbors to put up a fence on your own property..just the local zoning and planning office as many places require permits and have guidelines (actually legal requirements) as far as height, construction, and setbacks required.

Unlike Smoothy, I read this to suggest that the fence is on OP's property. If so, the neighbor cannot allow the trees to damage OP' fence. If branches grow through a chain link fence, eventually they will damage it.

I don't exactly know what "the last tree will block all access to our back yards" means. As noted, if it is not on OP's property, that's not the neighbor's problem (unless OP has an easement for such access).

This needs more description. Perhaps the fence is only a small part of your back yard? Or it is on the border between your yard and a common area?
Also - what country? It has a British sound.
Trees on borderlines are often matters of contention, and some laws are state or province wide, and some supercede those with local ordinances.
I would check with your local zoning board.

There, is normally not a common area, and you would not have needed permission to put up a fence.

If the fence is on the property line, then one side of the fence is yours, one side is theirs. If the other side is their property, they may plant trees there, if they wish, and they may do what ever they want, including not maintaining it.

If the other side of the fence is your property (why would you mow an area , not your property) then you can have the trees removed from your property.

If the trees are on their property, you have no rights to go onto their property without specific permission. And can not do anything to the trees, but can cut any limbs that come though the fence or goes over the fence.