<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Alex starts with a rooted tree with one vertex (the root). For a vertex  $v$ , let the size of the subtree of  $v$  be  $S(v)$ . Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex  $v_1$ ). The expected value of  $S(v_1)$  is of the form  $\tfrac{m}{n}$  for relatively prime positive integers  $m, n$ . Find  $m+n$ .**Note:** In a rooted tree, the subtree of  $v$  consists of its indirect or direct descendants (including  $v$  itself). Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
9901
Alex starts with a rooted tree with one vertex (the root). For a vertex  $v$ , let the size of the subtree of  $v$  be  $S(v)$ . Alex plays a game that lasts nine turns. At each turn, he randomly selects a vertex in the tree, and adds a child vertex to that vertex. After nine turns, he has ten total vertices. Alex selects one of these vertices at random (call the vertex  $v_1$ ). The expected value of  $S(v_1)$  is of the form  $\tfrac{m}{n}$  for relatively prime positive integers  $m, n$ . Find  $m+n$ .**Note:** In a rooted tree, the subtree of  $v$  consists of its indirect or direct descendants (including  $v$  itself).
test