The word “branch” means that which “shoots up,” or “sprouts” from the root of a tree, or from a decayed tree. The word ‘branch’ occurs six times in the Old Testament, see for example, Isa 4:2; Isa 11:1; Jer 23:5;  Jer 33:15; Zec 3:8; Zec 6:12. In all of these places, there can be no doubt that there is reference to Him who was “to spring up” from David, as a sprout does from a decayed and fallen tree, and who is, therefore, called a “root,” a “branch”, of the royal stock.

It is worth noting that the Lord is not described as a stately tree but as a shoot from the roots. We see this in practice when the Lord came to earth. At that time the family of David had lost all claims to the throne of Israel; it was an era of national ruin. Solomon’s side of David’s family carried a curse which disqualified their descendants from occupying the throne. Jeconiah was the last king of Judah from Solomon’s line (see Jer 22:30; Jer 36:30). Joseph, Mary’s husband, was a descendant of Solomon, Mat 1:6, but Mary of David’s son Nathan, Luk 3:31. No curse hindered her from being the mother of Israel’s king. The great tree of the Davidic monarchy had fallen, lying like a dead thing, but it was in this situation, as from the decayed root of a fallen tree, a “sprout” or “branch” came forth with more than the magnificence of David, who will succeed him on the throne. The name “branch”, is synonymous then with the title ‘the son of David’, or the Messiah.

Christ is called "the branch", not as God, but as a Man; it is a title indicating His descent from David. A branch or twig, or “shoot” is a tender thing, easily damaged, reminding us of Him growing up as a tender plant before the Lord, tender in His meekness, and full of compassion. At His first coming that little shoot seemed to be too fragile to have any power. However when He comes to earth again, the world will see what came from that little shoot, the beauty of His character will be on universal display.

The Schofield Reference Bible gives the following  comments with regards to this name of Christ, which is used in a four-fold way. He is referred to as:

(1) "The Branch of Jehovah" (Isa 4:2), that is, the "Immanuel – God with us" character of Christ (Isa 7:14) to be fully manifested to restored and converted Israel after His return in divine glory (Mat 25:31).

(2) The "Branch of David" (Isa 11:1); (Jer 23:5); (Jer 33:15) that  is, the Messiah, "of the seed of David according to the flesh" (Rom 1:3) revealed in His earthly glory as King of kings, and Lord of lords.

(3) “Jehovah's Servant, the Branch" (Zec 3:8).  Messiah's humiliation and obedience unto death (Isa 52:13-15; Isa 53:1-12; Phi 2:5-8).

(4) The "man whose name is the Branch" (Zec 6:12-13), that is His character as Son of man, the "last Adam," the "second Man" (1Co 15:45-47) reigning, as Priest-King, over the earth in the dominion given to and lost by the first Adam.

Matthew is the Gospel of the "Branch of David"; Mark is the Gospel of "Jehovah's Servant, the Branch"; Luke is the Gospel of "the man whose name is the Branch"; John is the Gospel of "the Branch of Jehovah."