The Cayley graph of the free group on two generators is the tree in which every vertex has four neighbours

Given: The objects and hypotheses in the Example.

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set $\{\{g,gs\}:g\in G,s\in (S\cup S^{-1})\setminus \{e\}\}$ (The Cayley graph of a group with respect to a subset).

The Cayley graph of a free group with respect to a free basis is a tree (The Cayley graph of a free group with respect to a free basis is a tree).

Every vertex of a Cayley graph has the same degree, and the graph is locally finite exactly when the symmetrised generating set is finite (Cayley-graph neighbourhoods are equipotent, and local finiteness is equivalent to finiteness of the symmetrised subset).

A cycle is a closed walk of length at least three with distinct vertices apart from its endpoints; a forest is a simple graph with no cycle and a tree is a connected forest (Cycles, trees and forests in a simple graph on an arbitrary vertex set).

A free group on a set $X$ is a group $F(X)$ together with a map $i:X\to F(X)$ such that, for every group $G$ and every function $u:X\to G$, there is a unique group homomorphism $\widehat{u}:F(X)\to G$ satisfying (Free group on a set of generators).

The subset $B$ is a free basis of $F$ if $(F,i)$ is a free group on the set $B$ in the sense of. (A free basis of a group).