If we take a graph invariant to be "a property that depends only on the abstract structure, not on graph representations such as particular labellings or drawings of the graph" (from Wikipedia), I have the feeling that Harrsion's question for complete graph invariants remained basically unanswered, since Greg's answer is mainly about (canonical) labellings.

Thinking - as Harrison did - of "the usual ones (the Tutte polynomial, the spectrum, whatever)", I'd like to repeat Harrison's question, but restrict it to trees:

Are there any known complete tree invariants?