Trees don't really "dry out" soggy areas. You'd do better to come up with a real drainage solution. Contact a landscape architect for an assessment of the situation.

I really feel you need to speak to your zoning and planning commision to let them know this is happening. Building contracters of developments may not be handling the run off properly and that may be what is causing your problem.
tell them what conditions you had in your yard before, if you didnt have this soggy situation then your right, its the increased building or septic systems from new houses leaching which would effect wells and others peoples drainage

you need to speak up.

I have lived in Westchester on and off for 40 years. The amount of development since the 1970's is astounding. Your solution will probably be part engineering and maybe part planting. Some plants like a wet area but I doubt they will dry it out to your satisfaction. Some plants that like it wet are: Willows (please do not plant weeping willow) and Red Osier Dogwood and some of the viburnums. You do need to check out what the source of the water is.
Good luck.

I had a wet area and planted a river birch. It turned out to be a great choice. Beautiful year round bark and the thing sucks up all water. (The area use to be a bit soggy and now it is not.) Your problem may not be the same, so I'd heed the advise of the others.

It is possible that your neighbors are running sump pumps to keep their basements dry. If you are downhill from the water runoff from those pumps, and just plain rain water, you get it all puddling in your yard. That is what happened to me at a previous home years ago. We ended up building a long berm of heavy clay soil and planted big cedar trees on top. Then the water started backing up in the nearest neighbors yard. It ended up that most of them dug and used dry wells to seep away their sump water. Worked for me! Water runoff on downhill properties can be a nightmare, as it seems to be a matter of sh** on your neighbor. Anyway, our berm not only stopped most/all of the water invasion, it also gave us a wonderful privacy wall. Good luck with your 'fix'. Pondy

In our townshipits against Zoning to have run off from your yard go into anyone elses. So although a berm might work, it might only be a temp fix for this person and if he "lies by" and doesnt do anything then at a later date if it becomes serious it may be to late.

Instead of a tree plant a rain garden.