Regular tree ($\mathrm{deg}(x)=k(x\in V)$)

An infinite connected tree is regular of degree $k$ if every vertex has exactly $k$ neighbours. Necessarily $k\ge 2$. For $k=2$ it is a two-sided infinite path. For $k\ge 3$, rooting it gives $k$ children at the root and $k-1$ children at each other vertex; this constructs the unique such tree up to graph isomorphism.