I was viewing the solutions of other Leetcode users for the classic "post-order traversal of a binary tree" question, when to my surprise, I found a ton of users simply finding the reverse preorder traversal (because it is considerably easier to implement iteratively), and then reversing the output. To my further surprise, I could not find a single counterexample to these conjectures, which I will state clearly:
Conjecture 1: The post-order traversal of a tree $T$ is equivalent to the reversed reverse pre-order traversal of $T$.
Conjecture 2: The reverse post-order traversal of a tree $T$ is equivalent to the reversed pre-order traversal of $T$.
I thought about this plenty, and came up with an extremely informal justification for this: LRN = reverse(NRL) and RLN = reverse(NLR). Not sure if this is purely coincidental or not. Can anybody either provide a proof of either conjecture? Furthermore, if true, do these conjectures extend to any arbitrary graph/digraph?