hey there for the broken branch , you had absolutely the right idea, kind of like when your own arm almost gets ripped up and you tappe it back together and it fixes itself no problem, only joking about that obviously out arms dont fix themselves like that. But the plant will actually fix itself very quickly by cauterising the wound in hours then starting to rebuild the fibers. Two things are not good for this process, light which burns the molecules and air which makes it "rust". So next time you want to fix it quicker use dark tape like the grey tape you can buy in a store called duct tape, main idea is no air not light, just wrap it in tape and the affected area will heal usually within a week or ten days (but i leave the tape on until the end just in case lol). Hope this helps ! ๐