Alma-2's words in Zarahemla
62 I speak by way of command unto you that belong to the church; and unto those who do not belong to the church I speak by way of invitation, saying: Come and be baptized unto repentance, that ye also may be partakers of the fruit of the tree of life. (Alma 5)
Notice how Alma-2 speaks by way of commandment to the members of the church (those who have made covenants). To those who have not, he speaks by way of invitation to become a partaker of the fruit of the tree of life. I hadn't noticed how clearly this verse connects covenant-making (and keeping) with the fruit of the tree of life.
Mormon's description of the Nephites after Christ's visit
3 And they had all things common among them; therefore there were not rich and poor, bond and free, but they were all made free, and partakers of the heavenly gift. (4 Nephi 1)
17 The fear of the Lord was upon all nations, so great was the glory of the Lord, which was upon his people. And the Lord blessed the land, and they were blessed upon the mountains, and upon the high places, and did flourish.
18 And the Lord called his people Zion, because they were of one heart and one mind, and dwelt in righteousness; and there was no poor among them. (Moses 7)
Moroni describes a purpose of Christ's visit to the earth
8 But because of the faith of men he has shown himself unto the world, and glorified the name of the Father, and prepared a way that thereby others might be partakers of the heavenly gift, that they might hope for those things which they have not seen. (Ether 12)
Here we have a clear mention of "a way" that was "prepar[ed]" by Christ in order to allow us to become "partakers of the heavenly gift."
This all fits in nicely with Nephi's vision which explains the meaning of his father's dream, in which a prophet is called to prepare the way for Him, He ministers in "power and great glory," twelve followers are carried away in the Spirit, and then angels descend upon the children of men and minister unto them:
27 And I looked and beheld the Redeemer of the world, of whom my father had spoken; and I also beheld the prophet who should prepare the way before him. And the Lamb of God went forth and was baptized of him; and after he was baptized, I beheld the heavens open, and the Holy Ghost come down out of heaven and abide upon him in the form of a dove.
28 And I beheld that he went forth ministering unto the people, in power and great glory; and the multitudes were gathered together to hear him; and I beheld that they cast him out from among them.
29 And I also beheld twelve others following him. And it came to pass that they were carried away in the Spirit from before my face, and I saw them not.
30 And it came to pass that the angel spake unto me again, saying: Look! And I looked, and I beheld the heavens open again, and I saw angels descending upon the children of men; and they did minister unto them. (1 Nephi 11)
This heavenly gift seems directly tied to partaking of the tree of life, and seems to be connected to the making and keeping of covenants. The process of being "taken up" and then sent back down as an angel to minister to others fits right into this symbolism. Just as Lehi partook of the fruit and immediately turned to offer it to his family.
What does it mean to partake of the fruit of the tree of life? I believe it is all about "covenant faithfulness." The Book of Mormon supports this idea.