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Consider this tree:

Simple binary tree

If I traverse it using post-order, I'd start at B (as it is the leftmost leaf) and that's where my misunderstanding begins. I know B is the first and A will be the last node in post order, as the rule is left-right-root. One of my university professors said the correct answer for the post-order traversal of a tree similar to the one above would be B, C, D, E, A, but in my understanding, it should be B, D, E, C, A.

Am I getting it wrong? Shouldn't I evaluate (C,D),(C,E) as a subtree and then go back to the parent tree?

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You are right. You finish each sub-tree before printing that sub-tree's root node. The correct order is B, D, E, C, A

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