I'm looking at the very first page of Raymond Smullyan's First-Order Logic and he talks about trees, defining an unordered tree $\mathcal{T}$ as a collection of the following
- A set $\mathcal{S}$ of elements called points.
- A function, $\mathcal{l}$, which assigns to each point $x$ a positive integer $\mathcal{l}(x)$ called the level of $x$.
- A relation $x\mathcal{R}y$ defined in $\mathcal{S}$ which we read as "$x$ is a predecesor of $y$" or "$y$ is a successor of $x$." This relation must obey the following conditions.
- $C_1$: This is a unique point $a_1$ of level $1$. This point we call the origin of the tree.
- $C_2$: Every point other than the origin has a unique predecessor.
- $C_3$: For any points $x$, $y$, if $y$ is the successor of $x$, then $\mathcal{l}(y) = \mathcal{l}(x) + 1$.
This seems to have a Peano's numbers feel to it. He goes on to further define trees. So I'm wondering where this came from? Does anyone know of another text that or resource that describes this tree? Is this parallel to anything else?