Consider a tree with k nodes and for each node v the vector **l**v = (lv0, lv1, ..., lvk-1) with lvd the number of leaves (!) with distance d to v. I wonder whether two nodes v, w with **l**v = **l**w are [conjugate](https://mathoverflow.net/questions/10807/conjugate-vertices-and-distinguishing-properties) (I guess they are). Can anyone help me to prove this - or give a counter example?



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.


