co.combinatorics - A distinguishing node property in trees?

I have a counterexample. It is not enough just to count leaves, since this doesn't take into account the number of possible ways to arrive at those leaves.

Consider the graph below.

   A - B - C - D - E - F
           |       |
           G       H
           |
           I

I think the vector for C and D both is 002200000, since they each have two leaves at distance 2 and two leaves at distance 3. But they are not conjugate, since C has degree 3 and D has degree 2.

I think this might be a minimal counterexample.