On the number of nonisomorphic subtrees of a tree
  We show that a tree of order $n$ has at most $O(5^{n/4})$ nonisomorphic subtrees, and that this bound is best possible. We also prove an analogous result for the number of nonisomorphic rooted subtrees of a rooted tree. 
\'Eva Czabarka, L\'aszl\'o A. Sz\'ekely, Stephan Wagner
math.CO