I'd dig a serious amount of grit and compost into that area to improve the drainage, after which it should be fine. One way to check drainage is to first dig a hole about 18" (45cm) deep, when it's forecast to be dry for a few days. If a puddle appears at the bottom after a day or two and doesn't drain away, then the ground is waterlogged and most trees won't like that at all. If that happens you can still improve things by digging a trench towards a lower part of the garden, filling the bottom with rocks, hardcore or gravel etc covered by a sheet of weed membrane before re-filling the trench with soil. Quite a lot of hard work but it will be a long-term fix.