Symbological: There are two different but connected images here. The first is partaking of the fruit of the tree of life, and the second is the eating and drinking of the bread and waters of life. Both of these images have root in the same symbols, but have probably taken different paths to Alma's expression.
It is most likely that Alma's imagery of partaking the fruit of the tree of life is related to Lehi's dream (see 1 Nephi 8:10-12). While there are no obvious restatements of the dream, certainly the dream of Lehi and subsequently Nephi would have continued to influence their progeny in the ages to come.
The second of the two images concerns the bread and waters of life. The most obvious reference for this imagery is found in the Gospel of John:
John 6:35 And Jesus said unto them, I am the bread of life: he that cometh to me shall never hunger; and he that believeth on me shall never thirst.
While the imagery of the water of life is not as explicit as the bread of life, nevertheless, the structure of the verse clearly implies the water. In the structure of John 6:35, we have the following logical pairings:
Bread of life: never hunger
(implied Water of life): never thirst
The structural pairing is very clear, and the "never thirst" must refer to a liquid rather than the "bread." Thus the Waters of Life are implied in that statement (the connection between Christ and the Waters of Life is made explicit in Revelations 21:6).
In spite of this usage in John, we need not require a New Testament source for this imagery in the Book of Mormon, as the New Testament imagery is dependent upon the Old World tradition of the Tree of Life - that tradition with which Lehi and Nephi would have been familiar prior to the newer vision of Lehi.
The symbology of the Tree of Life emphasized the dual aspects of the tree, the fruit of the tree itself, and the water that flowed from the base. Both of these were powerful to provide life to any who would partake of them (see, for instance, Ginzberg, Louis, The Legends of the Jews, (Philadelphia: Jewish Publication Society of America, 1909), v.1 p.93).
How do we know that these images come from the two differing sources, since both of them are related to a "tree of life"? The answer comes in the nature of what happens when one partakes of the fruit. In the case of the bread and water of life, those symbols imply life. Those who partake, according to Jesus as cited in John, will never hunger nor thirst. The effect is eternal.
On the other hand, those who partake of the fruit in Lehi's dream have no such promise of eternal benefit. For them, the taste of the fruit may be had, and subsequently lost (1 Nephi 8:28). This essential difference in the two images justifies the assertion that they come from different traditions.
That the tree of life here noted is specifically that of Lehi, where one may fall after tasting it, is not explicit, but rather is implied from both the text leading up to this statement, and the text which follows. Alma is speaking to members of the church who have fallen - who had once sung the song of redeeming love. He is calling them to repentance. It is specifically these to whom the repentance and ultimate acceptance of the Savior is directed. Thus this imagery of the fruit from Lehi's dream much better fits the context of Alma's sermon that the eternal aspects of partaking the fruit which were the tradition of the Old World Tree of Life.
Nevertheless, that imagery was also available, and in the context of the repentance of the sinner and acceptance by the Savior, this bread and water of life of which the repentant may partake also fits the symbolic context.