Last one! Not a single one of these has “do” in it, and half don’t even have a D. But they do come from do, the Proto Indo Europeanroot meaning “to give.” So there’s that.
The first few words are pretty obviously connected to each other. Betray showed up in the early thirteenth century, where it could also be bitrayen. The first part is from be-, which is just, you know, be, while the rest comes from tray, a Middle English word that isn’t used anymore (and has nothing to do with, like, a serving tray). Tray is from the Old French traine, from the verb trair, betray, from the classical Latin tradere, hand over. That word is also a mix of other words, and the tra- is from trans-, meaning across, and the rest is dare, to give, and that one’s from do-, making the word to give over. To betray is to be a give over-er.
Traitor’s origin is of course the same. It showed up in the thirteenth century from the Old French traitor/traitre, from the classical Latin traditor, a noun meaning traitor that’s just from tradere. Then there’s treason, which also showed up in the thirteenth century, from the Anglo French treson and Old French traison. That’s from the classical Latin traditionem, which actually means delivery and is also from tradere. And that traditionem looks an awful lot like tradition for a reason.
Tradition showed up later, in the late fourteenth century, from the Old French tradicion, which is just from traditionem. A tradition is something you “deliver” or hand down through the years. As for why it became treason in one word and tradition in another, well, another definition of traditionem is literally giving up/over. Which, yeah, could be a betrayal. It’s just so weird how differently you can interpret words.
The next word is a complete departure from the above. Add showed up in the late fourteenth century, from the classical Latin addere, to add. The ad- means to, and the rest is from dare, to give. To add is “to give to”. Makes sense!
Now for the last of the do- words, and perhaps the one that seems the weirdest: die. Wait, did you think I meant like to cease to live? Oh no. Not that one. I mean a die, like you roll in a game. The first die has its own origin. A die showed up in the early fourteenth century from the Old French de. Now, de is of “uncertain origin”, so no one’s sure, but it’s thought to be from datum, which I mentioned weeks ago as the origin for date and data. I have no idea why it would have anything to do with a die though, so maybe this speculation isn’t true. Still, weirder things about words have been true. I mean, who would think that add and traitor would be related?