Answer: There are several approaches to the problem, but unfortunately, planting a water loving tree probably would not solve it in the long run. I would strongly suggest that you obtain some expert professional advice about the drainage problem and how to correct it since water problems generally become worse over time. A landscape architect or other expert with professional training in surface grading should be able to figure out the best way to control and/or divert the runoff coming from the neighbor's property. It is usually possible to install an underground drainage system and possibly a sump pump to prevent water damage to the house. Also, in some areas it is illegal to direct runoff in such a way that it floods a neighboring property, so that may be worth investigating as well.
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