'Branches' in the Bible

they took branches of palm trees [in homage to Him as King] and went out to meet Him, and they began shouting and kept shouting “Hosanna! Blessed (celebrated, praised) is He who comes in the name of the Lord, even 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, for [otherwise] apart from Me [that is, cut off from vital union with Me] you can do nothing.

If anyone does not remain in Me, he is thrown out like a [broken off] branch, and withers and dies; and they gather such branches and throw them into the fire, and they are burned.