'Branches' in the Bible

So they took branches of palm trees and went out to meet him. They began to shout, "Hosanna! Blessed is the one who comes in the name of the Lord! Blessed is the king of Israel!"

"I am the vine; you are the branches. The one who remains in me -- and I in him -- bears much fruit, because apart from me you can accomplish nothing.

If anyone does not remain in me, he is thrown out like a branch, and dries up; and such branches are gathered up and thrown into the fire, and are burned up.