§ DFS numbers as a monotone map

Really, we want a partial order that is defined with the tree as the Hasse diagram. However, performing operations on this is hard. Hence, the DFS numbering is a good monotone map from this partial order to the naturals, which creates a total order. I want to think about this deeper, I feel that this might be a good way to think about the low numbers that show up in tarjan's algorithm for strongly connected components This also begs the question: can we use other partial orders, that chunk some information, but don't lose all the information as going to a total order (the naturals) does?