I had some problems with authorisation on a fairly new install of SourceTree, and decided to do an uninstall, cleanup, and reinstall to try to get back on track.
I followed instructions elsewhere to uninstall, remove Atlassian and Sourcetree folders in my C:\Users\<myuser>\Appdata\Local folder, and reboot and reinstallation. In addition, I removed all bitbucket related credentials in Windows Credential Manager.
Any attempt to start Sourcetree shows the app for a second or two and then disappears. Windows Event Log shows two errors a little similar to this ticket: https://confluence.atlassian.com/sourcetreekb/sourcetree-crashes-on-startup-831655339.html I tried solutions proposed there (though the user config file was not corrupted) and this did not work.
Having scanned the AppData\Local folder, I found this log entry in the SourceTree.log file:
ERROR [2020-06-23 07:51:21,458]  [SourceTree.Exceptions.RemoteHostException] [Log] - Insufficient authentication credentials
ERROR [2020-06-23 07:51:21,478]  [SourceTree.Exceptions.RemoteHostException] [Log] - Failed to list teams for user [XXXXXXXX_nij]
'XXXXXXXX_nij' is one of my bitcucket user names... BUT how does this installation know to search for that after all configuration folders have been cleaned up?
I had the exact same problem with those errors shown on the log file (with Sourcetree v3.8.9). My workaround was:
Hopefully, this will work for you too.
Those are basically the same steps that I followed; though I could not say if the version number I was trying to reinstall was v3.8.8!
Unfortunately I simply could not get the app running again on this particular computer; so I went to an alternative product called GitKraken (which I have now paid for the pro version).
There is a lesson for Atlassian here; I was happy to pay for a product that works well and has a reasonable amount of support -- I got NO support from Atlassian here in nearly two months and that is simply unworkable for me... this was not the first time that a reinstall or upgrade of Sourcetree left we scrabbling to fix some weird authentication or other issue; though on the whole I would have said in other respects Sourcetree was a very capable product.
I'm glad you got up and running!