Answer: In my experience mimosa trees tend to self seed all over the place and the easiest way to get a new plant is to transplant a tiny seedling in the late spring/early summer as they germinate. Take a huge shovel full of soil for a tiny plantlet just a few inches tall. That way it will not know it has been moved. The reason I suggest this is it is very difficult to grow these from seed in a container and transplant them successfully due to the sparse deep root system.
If you need to bring a seed pod home and can't dig up a seedling, wait until it is fully dry and mature on the tree and then set it outside where you want the tree to grow, in nature the seeds seem to germinate perfectly when they fall on old rotted down mulch so I would mimic that. Also, it is better to collect the seedpods from a tree close by if possible as it will be more cold tolerant that trees further south and thus the progeny will (hopefully) inherit that cold tolerance.
Good luck with your project!
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