All this is true.
Froth, in Shakespeare's Measure for Measure, II, 1, 565

Is it?

If a man could pass through paradise in a dream, and have a flower presented to him as a pledge that his soul had really been there, and if he found that flower in his hand when he awoke--Ay!--and what then?
Samuel Taylor Coleridge

Thanks to everyone here, I can now provide the following information about this tree. It's a Magnolia stellata, also known as a star or spider magnolia. The descriptions online mention shrubs or small trees, and it is slow growing, with pink or white flowers showing up before the leaves do each year. This must have been an old tree since it was relatively large. They can get up to 2.5m in height and 4.6 in width. Each flower has from 10 to 30 petals (or tepals, as they are called), depending on the cultivar. Oh, and they smell good! Thanks to everyone who identified this for me.