Why did God refer to Jesus Christ as a tree in Isaiah 11:1?

Jesus is the "shoot", not the "tree".

The main metaphor that Isaiah used is clearly that of a family tree:

• The "stump" is a metaphor for the family tree starting with Jesse, the father of David

• The "roots" of this family tree is living, producing many descendants, many branches, who were partially documented in the OT books (I have to look it up)

• The "Branch" is a metaphor for a single line of genealogy that NT author Matthew interpreted as the 29 generations between Jesse to Jesus in Matt 1:1-17, starting with Jesse in v. 6 until Jesus in v. 16.

• The "shoot" refers to the part of that Branch that will bear fruit. This "shoot" is the "him" in verse 2-3:

  The Spirit of the LORD will rest on him —
    the Spirit of wisdom and of understanding,
    the Spirit of counsel and of might,
    the Spirit of the knowledge and fear of the LORD—
  and he will delight in the fear of the LORD.
  

  Matthew and other NT writers identify this "shoot" to be a metaphor for Jesus, in whom the Holy Spirit rested giving him wisdom, understanding, fear of the LORD, etc. at his baptism (Matt 3:16, Mark 1:10, Luke 3:22).

Why is this important? NT writers taught that a portion (vv. 1-5) of the whole of Isaiah 11 prophecy had been fulfilled by God's providential preservation of the 29 generations from Jesse to Jesus (instead of lost such as among the Ten Lost Tribes) so that God can fulfill the covenant He made with David (1 Chron 17:11-14), hundreds of years before Isaiah, that one of his descendants will rule the kingdom forever. NT writers understood Jesus to be THIS person, the King of the everlasting kingdom (Act 13:22-23, Rom 15:12, Rev 5:5, Rev 22:16).

CONCLUSION: The parts of the tree mentioned in Isa 11:1 were a metaphor that Isaiah used to communicate a prophecy to Israel that God did NOT forget His covenant with David made hundreds of years earlier. NT writers continued that metaphor in identifying Jesus Christ as the "shoot", a physical descendant of Jesse, a fulfillment of the Davidic covenant that he will be the everlasting King promised to David.

P.S. (in light of your deleted question)

1. This tree metaphor should NOT be confused with another metaphor associated with Jesus in John 15:1-11 (Jesus as the true vine). A vine is not a tree. The metaphor was suggestive of a grapevine since v. 1 has the Father as the vinedresser who cuts fruitless vine branch and prunes a good vine branch so it bears more fruit (v. 2). This grapevine is NOT the same as the tree metaphor in Isa 11 since grapevines don't have a stump ! The function of this metaphor is also entirely different, since Gentiles are part of this grapevine but NOT the tree in Isa 11. The grapevine is a metaphor for union with Christ for all believers to draw sustenance from Him, etc. As believers we are related to Christ by adoption, not by blood represented by Isa 11's tree metaphor.
2. This tree metaphor should also NOT to be confused with the Olive tree in Romans 11 where Gentiles are grafted to Abraham's tree.
3. This tree metaphor should also NOT to be confused with the tree of life in the Garden of Eden, as the function is different than the generational tree from Jesse. If you would like to understand Jesus as this tree of life, then why not simply understand Jesus with multiple metaphors?

Well now, did he? According to Young's Literal Translation, the two verses in question read,

"And a rod hath come out of the stock of Jesse, and a branch from his roots is fruitful. Rested on him hath the Spirit of Jehovah. The spirit of wisdom and understanding. The spirit of counsel of might, the spirit of knowledge and fear of Jehovah" (Isaiah 11:1-2).

That speaks of the roots of Jesse, from which comes a branch and a rod. Certainly, the one spoken of in verse two makes us think of the promised Messiah, Jesus Christ, but nowhere is he spoken of as being a tree.

More prophetic words about trees appear in Jeremiah's prophecy, where he was instructed to write like this:

"And say to the forest of the south, Hear the word of the Lord... I will kindle a fire in thee, and it shall devour every green tree in thee, and every dry tree... Then said I, 'Ah Lord God! they say of me, Doth he not speak parables?' (Ezekiel 20:47-49 A.V.).

Interestingly, a most excellent comment about green trees came from Jesus himself; did he refer to himself as such a tree? Consider how he said this when on his way to be crucified:

"For if in the green tree they do these things - in the dry what may be done?" (Luke 23:31 YLT).

All those quoted verses are parables and not to be understood literally. The meaning becomes clear when we turn to what the apostle Peter said here:

"It is the time of the beginning of the judgment from the house of God, and if first from us, what the end of those disobedient to the good news of God? And if the righteous man is scarcely saved, the ungodly and sinner - where shall he appear? (1 Peter 4:17-18 YLT)

The reference to trees is to the stock from Jesse's root, the nation of Israel, out of which the rod of Christ appeared (Psalm 2:9). That nation tried to destroy him while he was yet in their midst, as one of them - thus the warning of worse to come when that green tree (the nation of Israel) became dry, i.e. ungodly and sinful, as stated in Psalm 89:32 referring to the seed (stock) of Jesse who would forsake God's ways:

"If they break my statutes, and keep not my commandments, then will I visit their transgression with the rod, and their iniquity with stripes." Which marries with the Messiah who "hath smitten earth with the rod of his mouth" (Isaiah 11:4 YLT).

My answer is that, in Isaiah 11:1, God speaks of the coming Messiah as being a rod, not a tree. Jesus applied "the green tree" parable to Israel, which would become dry and burned up, and he will be the instrumental rod with which it shall be struck. The Tree of Life in the Garden of Eden may well be a parable for Christ, but that's another questions!