I noticed a number of years ago that the "tree of life" as found in the New Jerusalem (Revelation 22:2) does not fit our assumptions. As you point out, it is in "the midst of the street" and "on either side of the river." I suppose one interconnected tree could fill the entire city. I can imagine the roots and branches joining in various places. But this would seem more like a jungle than a city.
Another possibility is that he is referring to the tree of life as a type of tree. We might say that we crossed the mountain and found that the fir tree filled the valley below. This would mean that the fir tree as a species filled the valley. There were fir trees everywhere we looked. This would make sense in the New Jerusalem. It measures 1500 miles wide, deep, and high. The people in it and using it must be a great multitude. Certainly, we do not expect millions of people to eat of the fruit of a solitary tree. Either the tree must be unusually large or there must be many trees. The wording of Revelation 22:2 points to many trees.
Your reference to Ezekiel 47:12 is very interesting. The river referred to here is the one that comes from under the temple altar and flows to the Dead Sea bringing life to this body of water. The trees shall grow on each side of the river (as in Revelation) and they shall bring forth as new fruit each month: "it shall bring forth new fruit according to his months" with "which bare twelve manner of fruits, and yielded her fruit every month" (Revelation 22:2). Also, in both cases, the leaves were for healing (Ezekiel says "medicine").
However, although the two verses seem to refer to the same tree, there are differences as well. Ezekiel certainly refers to the kingdom age--that is, the time of the millennial reign of Christ. However, Revelation 22:2 undoubtedly refers to the time of the new heaven and new earth--a time after the millennial reign. Also, the river in Ezekiel flows from under the altar in the kingdom temple down to the Dead Sea. The river in Revelation flows from the throne of God in the New Jerusalem that comes down from heaven (Revelation 22:1). Certainly, the two are patterned after one another, but they are not identical times of places.
You also mention the typology of the tree of life. This is not a settled fact. The Jews look on the tree of life as a symbol of the Torah (law of Moses) and immortality in the world to come. In the book of Proverbs, the tree of life is a symbol of wisdom 3:18), the fruit of the righteous (11:30), the desire fulfilled (13:12), and a wholesome tongue (15:4). In Genesis, it was a tree of which they could "eat, and live for ever" (Genesis 3:22). In Revelation, it is given to those who overcome (2:7) and to those who do God's commandments (22:14).
The tree of life has been used as a type of the word of God and as a type of Jesus Christ. There are certainly ways in which both can be taught. But no matter how we approach the tree of life, it is a picture of that which gives life and gives it more abundantly. It keeps showing up because God keeps showing up with His offer of life.
Your question about the tree of life in the Garden of Eden is intriguing. Why would they need a tree of life if there was no death in the world? And, how could there be death in the world until there was sin? But when sin entered the world, the tree of life was refused to those who needed it. It seems that the tree of life was only offered when it was not needed. Then, when the curse of death lay heavily on man because of his sin, the tree of life was removed as an option. Why?
I think that the answer is found in the true purpose of the tree of life. It was not meant to actually give eternal life to man. Rather, it was given alongside the tree of the knowledge of good and evil in order to show Adam and Eve that they had a choice to make. They must either choose life or death. When the law was given to Israel, we see a similar offer: "I call heaven and earth to record this day against you, that I have set before you life and death, blessing and cursing: therefore choose life, that both thou and thy seed may live" (Deuteronomy 30:19). The tree of life showed them that life came because they chose it. However, they could also choose death.
In similar, though somewhat reversed fashion, the tree of life is offered in the book of Revelation, not for the purpose of giving eternal life, but as a reward to those who have already chosen life. When Adam and Eve chose the forbidden fruit and reaped death, the tree of life was refused them. In Revelation, those who have already chosen life and have already overcome (Revelation 2:7) by believing in Jesus Christ (1 John 5:5) are given access to the tree of life. More than anything else, it is a picture of the offer
made to any in any age who has chosen God's way of salvation.