The tree pictured in many versions of the icon of the Nativity of Christ is a representation of the Jesse Tree.
The Jesse Tree is named after the Old Testament patriarch Jesse of Bethlehem, the father of King David.
Known as the Root of Jesse in the Greek tradition of Orthodox Christianity, the Root or Tree is a depiction of the genealogy or ancestry of Christ.
In Jewish prophecy the Messiah was supposed to be descended from David, and David himself is considered as a foreshadowing of the person of Jesus Christ. And so Jesse is shown as the root of this line of descent which culminates in Christ, who is depicted centrally within the tree as a child, sitting on the lap of His mother.
Another term for the Theotokos is the Rod of the Root of Jesse.
St Cosmas the Hymnographer composed hymns by weaving in Old Testament prophecies, showing the Virgin as the rod and the overshadowed mountain:
"Rod of the root of Jesse (Isaiah 11: 1), and flower that blossomed from his stem, O Christ, You have sprung from the Virgin. From the mountain overshadowed by the forest You have come (Habakkuk 3: 3), made flesh from her who knew not wedlock, O God, Who are not formed from matter."
The tree’s presence in the icon of the Nativity of Christ is to remind us of another fulfilled prophecy from Isaiah:
“A shoot shall sprout from the stump (tree) of Jesse and from his roots a bud shall blossom. The spirit of the Lord shall rest upon Him” (Isaiah 11: 1 - 2).
In the flesh, Jesus can trace His ancestry through both His mother and adoptive father Joseph, all the way back to Jesse.