MathDB

Alex starts with a rooted tree with one vertex (the root). For a vertex

v

, let the size of the subtree of

v

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).
Proposed by Yang Liu

Problem Statement