Robert, I did a Google search on umeboshi and found several links. This one was helpful:
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lists the tree as prunus mume; my reference book lists the common name as Japanese apricot, hardiness zones 6-9.
Another link that offers information and recipes is:
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site has been automatically translated into English and is somewhat awkward to follow the imperfect translation.)
This link has less info on the process of making umeboshi, but has a lot of info on the benefits of consuming it:
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was very interesting. I wasn't aware that the prunus genus included so many other fruits.
Here's a .pdf doc from the USDA on prunus mume that has a zone map and many more details:
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are several other links from a Google search. I'll let you go from there.
Good luck! SF