In the Lord of the Rings, there are constant instances when the number three is represented. This continuity of the concept of three could make one think of this as having significance, at least for Tolkien. Why did he not choose four, two, or five as numbers that would be used continuously throughout the saga? Tolkien chose three. Obviously this is a reference to the triune nature of God in the Christian tradition. Examples are as follows: Three rings were given to the Elves, Three men established the kingdom of Gondor: Elendil, Isildur, and Anarion; There are three Elven kingdoms: Rivendell, Lothlorien, and Mirkwood; The Fellowship eventually becomes a unit of three: Aragorn, Legolas, and Gimli; Frodo, Sam, and Gollum become an independent group with their own journey; Merry, Pippin, and Treebeard have their own story; Three wizards are mentioned: Gandalf, Saruman, and Radagast; Frodo turns 33 at the beginning of the story and Bilbo turns 111, which is three numbers and adds up to three, and the list goes on and on.
The importance of this theme is such that when people and/or kingdoms are involved, they act as one. This is parallel to the idea of God as three persons, but one being. In the Lord of the Rings, Tolkien seems to be putting forth the notion that something whole can still be one even if it has distinguishable parts with distinct functions. The parts or "persons" still are there to function for the existence of each other, which ultimately comprise the one. In a sense Tolkien is conveying that God is a relational being, in that His internal relationship should be the basis for how we should treat one another. The relationship between Aragorn, Legolas, and Gimli is that of working together in harmony and creating a sense of interconnectedness in order to achieve the same collective goal. Intuitively, they can only do this if they act as one unit. Gandalf seems to be the only character who brings an "awakening" light to a couple of the triune groups such as Aragorn, Legolas, and Gimli, and Merry, Pippin, and Treebeard. This can be an allusion to how Christ brought the reality of the Trinity to humankind through his death and resurrection.