Happy is the man that findeth wisdom... she is a tree of life to them that lay hold upon her: and happy is every one that retaineth her(It's hard to avoid the interesting side-note of the divine feminine, summarized in this post by Lord Wilmore). But most modern Jews interpret wisdom to be the Torah so they make a connection between the Torah and the Tree of life based on this verse. Another basis for this connection is related to the cherubim who guard the tree of life. Later we find them on the Ark of the Covenant (see Exodus 25:17-22). Inside, the ancient Israelites had the stone tablets, God's law, i.e. the Torah. So again, Jews are making this connection between the Torah and the Tree of life.
I think the Jews are on to something here, but I think the Book of Mormon helps deepen our understanding. I would generalize and say that wisdom or the stone tablets represent the word of God. For ancient Jews, that was the Torah. For us, it is the scriptures, the living prophets and the guidance of the Holy Ghost. These three are mentioned in footnote 4 in this talk from Elder Neil L. Andersen from last General Conference about fruit. He relates this to the iron rod that we read about in the Book of Mormon. As we know, the iron rod leads to the tree of life.
It is interesting to note that the two accounts about a process leading to the tree of life in the Book of Mormon, 1 Nephi 8 and Alma 32, both describe the word of God as the key ingredient to get there. In Lehi's dream, the word of God is the iron rod that leads us to the tree of life. In Alma 32, the word of God is compared to a seed. As we nourish it, it will grow to a tree of life (Alma 32:40, Alma 33:23). The Torah or the word of God is not necessarily itself the tree of life, but it is the means to get us there. When we study the word of God, we find wisdom, which is, as the passage in Proverbs stated, a tree of life.
Perhaps the cherubim that prevented Adam/mankind from partaking of the fruit of the tree of life were placed on the Ark of the Covenant to point to the stone tablets inside. It's a gesture suggesting that we cannot take part of the tree of life right now, but we need to pay attention to the stone tablets, the law, the Torah, the word of God. Because if we cling to it like an iron rod or plant it in our hearts and nourish it, then one day we will be able to return to the tree of life and partake of the fruit at last. The word of God will get us there.