from klao, "to break" (cp. klasma, "a broken piece"), properly a young tender shoot, "broken off" for grafting, is used for any kind of branch, Matthew 13:32; 21:8; 24:32; Mark 4:32; 13:28; Luke 13:19; the descendants of Israel, Romans 11:16-19,21 .
akin to klao, "to break," denotes "a tender, flexible branch, especially the shoot of a vine, a vine sprout," John 15:2,4-6 .
from steibo, "to tread on," primarily denoted "a layer of leaves, reeds, twigs or straw, serving for a bed;" then "a branch full of leaves, soft foliage," which might be used in making a bed, or for treading upon, Mark 11:8 .
of Egyptian origin, frequent in the papyri writings, denotes "a branch of the palm tree," John 12:13 .
Note: Matthew, Mark and John each use a different word for "branch" in narrating Christ's entry into Jerusalem.