Hausdorff Maximal Principle "Hausdorff's Maximal Principle" says that any partial order P has a maximal chain (chain = linear suborder). It is equivalent to the axiom of choice. If we restrict Hausdorff's Maximal Principle to trees, i.e. partial orders, where a minimum element exists, and the sets $\\{x \mid x < y\\}$ form linear orders, what exactly do we get? Is it still equivalent to AC?

Even restricting this to well founded trees is enough to get $\sf DC_\kappa$ for every $\kappa$, which is enough to prove the axiom of choice.

So the answer is indeed positive.