A tree which each node selected as a root results in a different topology?
Here is a hint. If the tree has size $&gt; 1$ it will have to have a vertex with degree $\ge 3$ as otherwise it would be in a line and the two ends would be equivalent. So consider a tree with a vertex having 3 branches out of it. The branches will have to have different lengths or the end vertices of equal length branches would be equivalent.

What's the smallest number of vertices in a tree of that form? Can you see how to expand the tree maintaining the required property?



Also note that this approach proves that the sizes of trees having the mentioned property are 1 and $\ge 7$.