WikiTree currently serves profiles with the XHTML 1.0 Transitional doctype tag.
The XHTML specification requires <br /> , and the <br> form is not allowed. So, that is your "technically correct" answer.
Browsers typically allow incorrect code in pages (otherwise people complain and use a different browser), so the "practical" answer is that they are all interchangeable and will be for the foreseeable future.
What about the future though? I can't comment on any direction WikiTree might move in the future ; whether they start enforcing XHTML 1.0 Transitional, or whether they might change specification. So the "technically correct" answer may change over time, but the "practical" answer stays the same; I do not foresee any of the three forms ever being rejected by browsers.
I tried a random WikiTree profile in the W3C validator for XHTML 1.0 Transitional, and it reported 200 errors and 158 warnings; so at this point in time, WikiTree is not making an effort to actually follow that specification itself, nor have profiles follow it.