Infinite Trees, Infinite Branches, And Choice 
https://en.wikipedia.org/wiki/Tree_(set_theory)

Defines a tree T order-theoretically as a poset that has a minimum element R, and for each t in T, the set of lower bounds of T is well-ordered. This makes sense. 
If I take the above definition and say "an infinite tree possesses an infinite branch", this is some kind of choice principle, right? 
How does this relate to countable choice, dependent choice, and the axiom of choice? Is this principle something like dependent choice? 
 A: There are a few unclear things with your question. What is an infinite tree? If you have just one level above the minimum, is the tree infinite? So we need to talk about height, the least ordinal strictly larger than all the levels of the tree. But even then we have a problem, since we can write down an example of a tree which has height $\omega$, but every node has finitely many successors, so there are no infinite branches.
Therefore we really want to say something like "a tree without leaves" (which will immediately imply the height is an infinite ordinal). And now we can actually talk about what guarantees a branch.
Dependent Choice is equivalent to the statement "Every tree without leaves has an infinite branch". In the one direction, just use Dependent Choice to produce a branch, the same as you would done in the usual case of $\sf ZFC$. In the other direction, if we have a non-empty set with a relation on it which satisfy the assumptions of Dependent Choice, then the finite sequences which satisfy the conclusion (i.e., consecutive nodes are related) form a tree without leaves, and a branch is a function witnessing the conclusion.
Note, however, that the above depends on how you define a branch. If you define a branch as a maximal chain, rather than just a chain, then "Every tree without nodes has a branch" is in fact equivalent to the axiom of choice, as you can work out a proof that for every $\kappa$, Dependent Choice for sequences of length $\kappa$ holds. Which in turn imply the axiom of choice.
