example of tree (set theoretic)

The set ${\mathbb{Z}}^{+}$ is a tree with . This isn’t a very interesting tree, since it simply consists of a line of nodes. However note that the height is $\omega$ even though no particular node has that height.

A more interesting tree using ${\mathbb{Z}}^{+}$ defines if $i^a=m$ and $i^b=n$ for some $i,a,b\in {\mathbb{Z}}^{+}\cup \{0\}$. Then $1$ is the root, and all numbers which are not powers of another number are in $T_1$. Then all squares (which are not also fourth powers) for $T_2$, and so on.

To illustrate the concept of a cofinal branch, observe that for any limit ordinal $\kappa$ we can construct a $\kappa$-tree which has no cofinal branches. We let and . The tree then has $\kappa$ disjoint branches, each consisting of the set for some . No branch is cofinal, since each branch is capped at $\beta$ elements, but for any , there is a branch of height $\gamma +1$. Hence the supremum of the heights is $\kappa$.