Sufficient condition for root reconstruction by parsimony on binary trees with general weights

We consider the problem of inferring an ancestral state from observations at the leaves of a tree, assuming the state evolves along the tree according to a two-state symmetric Markov process. We establish a general branching rate condition under which maximum parsimony, a common reconstruction method requiring only the knowledge of the tree topology (but not of the substitution rates or other parameters), succeeds better than random guessing uniformly in the depth of the tree. We thereby generalize previous results of [13, 37]. Our results apply to both deterministic and i.i.d. edge weights.