The Staff of Yehudah

The Ben Ish Chai, zt”l, explains why the verse only refers to the heads of the exile in Bavel as a shevet, or staff. He cites the Seder Hadoros:
Once there was a Parthian king who decided to eradicate the seed of Dovid in his area. He began killing everyone known to be a descendant of Dovid, sparing no one.
He was so ruthless that he managed to eradicate everyone except a single widow who was newly expecting a child from her brutally murdered husband. But of course the king’s next step was to track down all of the wives of those he had killed to ensure that no child survive his decree.
That very night he dreamed that he was in a resplendent garden with magnificent tree bearing all sorts of luscious fruits. But the king was in a angry mood in the dream and he began to chop down all the tree of the garden until none remain aside from a little sapling, standing straight as a shevet, or staff. The king lifted up his axe to destroy this final remnant when an old man with ruddy skin and beautiful eyes gave a great shout, grabbed the axe from the king and struck him a blow to the head.
The king fell at the feet of the old man begging him to spare his life. “Isn’t it enough that you destroyed my garden, you must also destroy this little shoot that stands like a staff planted in the garden?”
The king begged forgiveness and swore to change his ways. If the old warrior would spare him he would devote his days to protecting this shoot and nourishing it. He swore solemnly to care for it always, day and night, planting from its fruit until the entire garden is restored.”
The old man spared him and he woke up terrified.
He called the Jewish sages into the room and they explained that the old man had been Dovid Hamelech and that he had sworn to protect his descendant.
The king was as good as his word and made a special place in his palace for the young mother to be, giving her everything she required. And this kindness extended to the child she bore, who was named Rav Bustenai after the garden or “bustan,” in the king’s dream.
The Ben Ish Chai concluded, “This is why the verse uses the word שבט to allude to the exilarchs from the line of Bustenai.”