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 $\frac{m}{n}$ 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).

Let's think step by step and output the final answer within \boxed{}.
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