Recall that rooted trees may be generated by starting with a trivial rooted tree (just a vertex), along with the operations of grafting a number of trees (identify their roots) and adding a new vertex to the tree to be a new minimum element. We will call this second operation "leafing".

Now let us define an invariant of rooted trees. If $T$ is a rooted tree, we will denote $P_T(z)$ to be the associated polynomial.

If the number of edges of $T$ is zero, then $P_T(z)=1$.

If $T'$ is the leafing of $T$, then $P_{T'}(z)=(z+1)P(z)+1$.

If $T$ is the grafting of $T_i, i=1\ldots n$, then $P_T(z)=P_{T_1}(z)P_{T_2}(z)\ldots P_{T_n}(z)$.

This polynomial is an isomorphism invariant of rooted trees. My question is

If $P_T=P_{T'}$, are the rooted trees, $T,T"$ isomorphic? If these trees are not isomorphic, what is the smallest counterexample? Any references to this invariant would be appreciated.