Since there seems to be some confusion in the comments below Richard Stanley's answer, and maybe also some discrepancy in terminology between Owen's answer and Richard's, I will record what I think is going on.
Vertices of rooted trees can be ordered by $x \lt y$ if $x$ is a descendant of $y$. The notion of subtree used by Owen looks as though he means upward closed subsets, since his subtrees include the root of the original tree (I apologize to Owen if that's not what he meant, although I think it is because that seems to be consistent with his remark on coefficients).
But that's not the usual notion of subtree, which according to Wikipedia is a (principal) downward closed subset of the original tree (i.e., if $y$ belongs to the subtree and $x$ is a descendant of $y$, then $x$ belongs to the subtree). Under that notion, Richard's answer made a lot more sense. Let me describe what I think the isomorphism types of his examples are using ZF sets. Let $a, b, c, d$ be ur-elements, and order sets by the transitive closure of the membership relation (so that $x \in y$ implies $x \lt y$). Then I can guess one of his trees looks like
$$\{ \{a, b, c\}, \{ \{ d \} \} \}$$
which has four one-node subtrees $a, b, c, d$, one two-node subtree $\{d\}$, one three-node subtree $\{\{ d \} \}$, one four-node subtree $\{a, b, c\}$, and one eight-node subtree which is the original tree (N.B. here, $n$-node subtree means there are $n$ vertices, including its root.) The other of his trees looks like
$$\{ \{a, \{ b \} \}, \{c, d \} \}$$
which has four one-node subtrees $a, b, c, d$, one two-node subtree $\{ b \}$, one three-node subtree $\{c, d \}$, one four-node subtree $\{a, \{ b \} \}$, and one eight-node subtree which is the original tree. (If those were not the isomorphism types he had in mind, then again I apologize.)
I don't believe however that these two trees have the same polynomial. If I did my arithmetic correctly, I believe they have different constant coefficients (one is 35 and the other is 36), using the original definition of the polynomial, not Owen's modification.
