Extremal first passage times for trees

For random walks associated with trees with probability zero of staying at any vertex, we develop explicit graph theoretic formulas for the mean first passage times between states, we give lower and upper bounds for the entries of the mean first passage matrix E, and we characterize the cases of equality in these bounds. We also consider the variance of the first return time to a state and we find those trees which maximize the variance and those trees which minimize the variance. As may be expected, the trees which provide extremal behavior are given by paths and stars.