Overhanging Tree Dispute with Neighbor

[Jeremiah Williams]

The neighbor of one of my rental properties contacted us about a tree between our two houses that has limbs overhanging both properties. The entire trunk of the tree is on the neighbor's property. The neighbor is asking us to split tree trimming costs with him, which doesn't seem to make sense. 

With the tree trunk being on his property, wouldn't he be solely responsible for the costs of maintenance?

Has anyone been through a similar experience?