  An irreducible stochastic matrix with rational entries has a stationary
distribution given by a vector of rational numbers. We give an upper bound on
the lowest common denominator of the entries of this vector. Bounds of this
kind are used to study the complexity of algorithms for solving stochastic mean
payoff games. They are usually derived using the Hadamard inequality, but this
leads to suboptimal results. We replace the Hadamard inequality with the Markov
chain tree formula in order to obtain optimal bounds. We also adapt our
approach to obtain bounds on the absorption probabilities of finite Markov
chains and on the gains and bias vectors of Markov chains with rewards.
