Our tree?

Recently my neighbor's tree, which crosses onto my property, has been making a mess of my pool. The branches and leaves, which are above my pool, constantly fall all over damaging my recently fixed pool and yard. How do I go about politely telling my neighbor of my problem and what can/should I do about it. If the decision to prim/cut the branches came about, would we both split the bill or would I solely be responsible? This is a first for me since my neighbors are new to the neighborhood and this is the first time that the tree has ever made such a mess.