 Another representation of the free monoid on \(n\) generators is as an infinite tree, every vertex of which has \(n\) descendants, one for each letter of the alphabet, with each edge labeled by the corresponding letter. Each vertex \(v\) represents the word consisting of the letters encountered along the path from the root to \(v\). The concatenation of \(u\) and \(v\) is the vertex arrived at by taking the subtree whose root is the vertex \(u\), noticing that this tree is isomorphic to the full tree, and locating \(v\) in this subtree as though it were the full tree.