  We use random destruction processes such as the random cutting model or site
percolation on rooted trees to define several polynomial invariants for rooted
trees. Some properties of these invariants are then exhibited, including
recursion formulas and relations between them. The main result states that two
of these invariants are complete, that is they distinguish rooted trees (in
fact, even rooted forests) up to isomorphism. The proof method relies on
recursion formulas and on irreducibility of the polynomials in suitable unique
factorization domains. For other invariants, we provide counterexamples showing
that they are not complete, although that question remains open for a certain
trivariate invariant.
