Hechler proved that you can force the existence of a cofinal subset of $\omega^\omega$ having any shape you like (subject to one or two necessary restrictions):

**Theorem:** (Hechler) Suppose that $(P,\leq)$ is a partially ordered set with the property that every countable subset of $P$ has a strict upper bound in $P$. Then there is a ccc forcing notion which adds a cofinal subset of $(\omega^\omega,\leq^*)$ that is order isomorphic to $(P,\leq)$.

The result can be found in:

*Hechler, Stephen H.*, On the existence of certain cofinal subsets of ${}^\omega\omega$, Axiom. Set Theor., Proc. Symp. Los Angeles 1967, 155-173 (1974). ZBL0326.02047.

**EDIT:**

In my original answer to this question, I said that Hechler's result implies that, yes, any reasonable-looking tree can be the order type of a cofinal subset of $\omega^\omega$. K. P. Hart pointed out to me in an email that this is not in fact the case.

The poset $(\omega^\omega,\leq^*)$ is (countably) directed, which means that it cannot contain two incompatible elements. The same is true for any cofinal subset of this order. Thus the only trees that are allowed as the order type of cofinal subsets of $(\omega^\omega,\leq^*)$ are the unbranching trees with uncountable cofinality: i.e., the type of $\mathfrak{d}$ when $\mathfrak{b} = \mathfrak{d}$. (Or at least the type of some ordinal $\kappa$ with cofinality $\mathfrak{d}$, which can be attained by packing extra functions into a scale in certain models of $\mathfrak{c} > \mathfrak{d} = \mathfrak{b}$.)

But that being said, if $(T,\leq)$ is any tree-like poset such that every branch has uncountable cofinality, then $T$ can be used to generated a countably directed poset $(P,\leq)$, namely the poset of downward-closed subsets of $T$ that do not contain a branch, ordered by inclusion. (I'm envisioning an upward-growing tree when I say "downward closed" here -- feel free to reverse it according to your own preferences.) And this poset can, in a Hechler-style forcing extension, be isomorphic to the order type of a cofinal subset of $(\omega^\omega,\leq^*)$.