In set-theoretic trees, we have the binary relation $<_T$ (which defines the "ancestor-descendant" ordering). This relation is partial, as children are incomparable in that ordering.
Could someone point me in the right direction to find the theory behind trees for which we also have an order between siblings that share the same parent? Here, we would have the first, second, ... $n$-th child of a given node in the tree.
(I am new to set or order theory. Excuse me if this is a trivial question, but I couldn't find an answer to something that sounds so trivial.)
