Falling Zero

You are given a tree of $N$ nodes rooted at node $1$. Each node $u$ initially has a positive value $a_u$ associated with it. You randomly choose one node in the tree and change its value to zero. If any node $u$ has a value equal to $0$, a random node $v$ will be chosen in the subtree of $u$ ($u \neq v$), and the values $a_u$ and $a_v$ will be swapped, hence the value of the node $v$ will becom