I am not sure if this exactly what you want, but here is a theorem that might help:

**Definition** Assume $\mu\le\kappa\le\lambda$ and $\mu$ is a regular cardinal. (1) Let $D(\kappa,\lambda)$ iff there is a linear order of size $\lambda$ and a dense set of size $\kappa$. (2) Let $D(\kappa,\lambda,\mu)$ iff there is a linear order of size $\lambda$, character $\mu$ that has a dense subset of size $\kappa$.

Then

**Theorem** (1) $D(\kappa,\lambda,\mu)$ iff there is a tree of height $\mu$, cardinality $\le\kappa$ and at least $\lambda$ branches of length $\mu$. (2) $D(\kappa,\lambda)$ iff there is a tree of height $\le\kappa$, cardinality $\le\kappa$ and at least $\lambda$ branches.

This is from J.E. Baumgartner, *Almost-disjoint sets the dense set problem and the partition calculus*, Annals of Mathematical Logic, Volume 9, Issue 4, May 1976, Pages 401-439