When working with phylogenetic networks, how does one talk about reticulate (i.e., web-like) the branches are? Is there a standard measure of this? If so, can someone recommend an R package that can calculate it?

A tree can't be web-like since any two nodes are connected by only one path.
Maybe just counting the number of internal nodes is what you're after.
What is the question you're trying to address with this ?

Sorry, I was thrown off by the previous comment. I am not familiar with phylogenetic networks. I would think that various concepts from graph theory could be useful but which to choose would depend on what you're trying to achieve.

Can you clarify what you mean by "how web-like" a tree is?

A tree can't be web-like since any two nodes are connected by only one path. Maybe just counting the number of internal nodes is what you're after. What is the question you're trying to address with this ?

Not phylogenetic trees, phylogenetic networks. By "web-like" I mean the horizontal lines between branches.

Sorry, I was thrown off by the previous comment. I am not familiar with phylogenetic networks. I would think that various concepts from graph theory could be useful but which to choose would depend on what you're trying to achieve.