The set $\mathbb{Z}^+$ is a tree with $<_T=<$ 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 $m<_T n$ 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 $T=\{(\alpha,\beta)|\alpha<\beta<\kappa\}$ and $(\alpha_1,\beta_1)<_T(\alpha_2,\beta_2)\leftrightarrow \alpha_1<\alpha_2 \wedge \beta_1=\beta_2$ The tree then has $\kappa$ disjoint branches, each consisting of the set $\{(\alpha,\beta)|\alpha<\beta\}$ for some $\beta<\kappa$ No branch is cofinal, since each branch is capped at $\beta$ elements, but for any $\gamma<\kappa$ there is a branch of height $\gamma+1$ Hence the supremum of the heights is $\kappa$