How to go from Tree to Total orders

Question

Given a tree $T=(X,E)$, is it guaranteed for any orientation of the edges $E$, there exist a strict total order preserves it?

For instance, let $X=\{x_1,x_2,..x_n\}$ and $E=(x_i,x_{i+1})$ the result is a tree $T$. Let $G$ be directed graph of $T$ (add directions into the set of edges in $T$). is it always the case where there is a total order $\succ$ extend/equals $G$?

Answer 1

Yes, by induction on the number of nodes. It is clearly true if there is just one node. In general, remove a leaf node, order the remaining tree, and insert the leaf node in the order. There is only a single constraint to be satisfied in doing so, and it is trivially satisfied.

Answer 2

Yes, every order can be extended to a total order by the Szilprajn extension theorem.