HasChildren could have been Parent, HasNoChildren could have been Leaf or Child. And so many more things.
Actually having a email client software running on your machine is extremely nitch and is mostly in the realm of self-hosters and legacy holdouts that won't let their clients go.
A most advanced modern approach is to just use POP3 to download your emails to a local Maildir and have them indexed there non-destructively. And then sync between your various machines that you want your email accessible using some sort of file sync or P2P solution.
I use notmuch for this. It automatically indexes and tags emails and thus enables much more advanced email management solutions then what can be offered over something like IMAP.
The main advantage of this is that 'folders' are managed virtually. There is no shuffling or copying or editing of emails done normally. I only have to worry about backing up my emails and notmuch config as all the rest can be regenerated relatively quickly.
This is more or less replicating what Gmail and other webmail providers do server side.
Where as the traditional approach shuffling and moving and deleting of emails on some imap server is fairly dangerous and expensive operation. Mistakes can lead to data loss and are often very difficult to reverse.