Answer: Based on your description I am not certain what the little trees would be. They could be a fast growing weed or they might actually be tree seedlings. Usually, mowing often and maintaining healthy lawn grass will crowd these out over time. You could also dig them out (roots and all) or spot treat with an herbicide containing glyphosate. (This is a non selective weed killer and will kill grass as well as weeds, you may find a wipe on application method gives you better control.)
Your lawn should continue to improve if cared for correctly. Set your mower blade at 3 inches and mow often enough that you remove no more than one third the grass height when you cut it.
Run some basic soil tests and check the fertility and pH of the soil. Fertilize and add lime as indicated by the soil tests.
Sandy soil can be improved by adding organic matter such as compost. This can be done by topdressing the lawn. Also, use a mulching blade on your mower and leave the grass clippings on the lawn as an additional source of both organic matter and nitrogen.
If you need to overseed, late summer to early September is the best time of year to do that. Seeding in July and August is usually futile -- the weather is too hot for the new grass to handle.
Your local Cornell county extension should be able to help you with the test results and interpreting the results. They should also be able to help you develop a lawn improvement and maintenance plan based on your soil tests and current condition of the lawn. Good luck with your project!
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