The Rod out of the Root of Jesse
Old Testament: Isaias 11:1
And there shall come forth a rod out of the root of Jesse, and a flower shall rise up out of his root.
Old Testament, also: Daniel 4:23
But whereas he commanded, that the stump of the roots thereof, that is, of the tree, should be left: thy kingdom shall remain to thee after thou shalt have known that power is from heaven.
New Testament: Romans 15:12; Apocalypse 22:16
And again Isaias saith: There shall be a root of Jesse; and he that shall rise up to rule the Gentiles, in him the Gentiles shall hope... I Jesus have sent my angel, to testify to you these things in the churches. I am the root and stock of David, the bright and morning star.
The connection
Isaias says root, not tree. The prophet is writing directly after a chapter that felled a forest, and a root is what an axe leaves. The house of David is not addressed as a standing dynasty but as the part below the cut, and the promise is that the part below the cut is where the rod comes from. Daniel supplies the same mechanism in a case where the reader is told outright what it means: cut down the tree, but leave the stump of the roots in the earth, and the kingdom remains to the man once he has known that power is from heaven. Humiliation there is not the end of the line, it is the condition of its restoration. Paul then quotes Isaias by name and takes the promise outward, so that the root of Jesse rises to rule the Gentiles and in him the Gentiles hope, and a shoot from a cut stump becomes the ground of hope for people who were never part of the tree. The Apocalypse closes it from the other direction. Christ does not call Himself the shoot but the root and stock of David, which reverses the order of nature, since He is the source of the line He grew out of.
Moral sense
What survives a felling is not what was tallest but what was lowest and least visible. A root is the part of a tree that was never admired.
Anagogical sense
The rod that came out of the root is finally named as the root itself, so the line that produced him turns out to have been growing out of him the whole time.
The Fathers
St Irenaeus, Against Heresies, Book III, ch. 9, sect. 3, applying the prophecy to Christ and to the Spirit resting on him: "There shall come forth a rod from the root of Jesse, and a flower shall rise from his root; and the Spirit of God shall rest upon Him."
Warrant
Romans 15:12 names Isaias and quotes him, so the New Testament supplies the identification itself, and Apocalypse 22:16 takes up the same root and stock of David. Daniel 4:23 is joined here as the passage that spells out the mechanism of a stump deliberately left in the ground, and stands as a parallel rather than as a text the New Testament cites. Checked in the Greek: rhiza is the same noun at Isaias 11:1 in the Septuagint and at both Romans 15:12 and Apocalypse 22:16. Paul's quotation at Romans 15:12 follows the Septuagint of Isaias 11:10, a few verses on from the rod itself, which is why the root and not the shoot is the word that carries into the New Testament. The name carries too: Iessai stands at Isaias 11:1 in the Septuagint and at Romans 15:12, so Paul keeps the root and the man it belongs to.
Catechism of the Catholic Church, paragraph 437.