What is an example of an infinite tree with depth 3?

Question

The example should be formalized as a partial order set. (A set and a relation over that set.)

I'm not sure how to go about this. Each layer, or at least one of the layers, has to have infinite successors, but there can only be 3 total layers of the tree. I really have no clue what direction to go here.

Answer 1

An even simpler answer: Consider $P=\mathbb Z_{>0}\times\mathbb Z_{\geq0}\cup\{(0,0)\}$ with the relations $(a,b)\leq(a,0)\leq(0,0)$ for all $(a,b)\in P$.

Answer 2

Consider the set $S = (\mathbb{Z} \times \{0\}) \cup (\{1\} \times \{1, 2\})$ with the relation $R = \{((a, b), (c, d)) \in S^2 \mid b < d \lor (b = d \land a = c)\}$.

Answer 3

Just go and built it. You want a tree, so there better be a root. Call it $r$. Then you want that the tree has depth $3$, so lets extend it to $r < u < v < w$. Alright, now we need to it be infinite without messing with the depth. So lets put in all $n \in \mathbb{N}$ as additional children of the root, i.e. add $r < n$ for each $n \in \mathbb{N}$. And we are done.