TITLE: Complete tree invariants?
QUESTION [7 upvotes]: 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?

REPLY [7 votes]: @Hans: The path sequence of $G$ is $(p_1, p_2, ...)$ where $p_i$ is the number of $i$-edge path subgraphs of $G$ isomorphic to an $i$-edge path.  For a tree, this is the same as the number of pairs of vertices at mutual distance $i$.  Morin, Wagner and I proved [arXiv:math/0609339] that the path and degree sequences of a tree (actually, something a but more general) can be determined from its chromatic symmetric function, but OTOH the path and degree sequences do not together determine the tree up to isomorphism; there is an 11-vertex counterexample in our article (due to Eisenstat and Gordon).