Full Path Eraser

There is a rooted tree of $N$ vertices rooted at vertex $1$. Each vertex $v$ has a value $A_v$ associated with it. You choose a vertex $v$ (possibly the root) from the tree and remove all vertices on the path from the root to the vertex $v$, also including $v$. This will result in a forest of zero or more connected components. The beauty of a connected component is the $\mathrm{GCD}$ of the valu