Given an undirected tree $G=(V,E)$, I want to decompose the tree into several subtrees. Let's denote the size of $i$th subtree by $|v_i| \, \forall \, i \in {1,2,...n}$. Note that the resulted decomposed tree has $n$ vertices, then. Also note that we also aim to determine the value of $n$ and it is not predefined. I want to obtain a decomposition that minimizes $$|v_1|! \times |v_2|! \times ... |v_n|! \times n!$$
I have a sparse tree with over 1000,000 vertices. I would appreciate if anyone can suggest an efficient algorithm or mathematical model to find such decomposition. For instance in this example, we have $2! \times 2! \times 2! \times 3!$.
