Birth and Death Chains on Finite Trees: Computing their Stationary Distribution and Hitting Times

Every birth and death chain on a finite tree can be represented as a random walk on the underlying tree endowed with appropriate conductances. We provide an algorithm that finds these conductances in linear time. Then, using the electric network approach, we find the values for the stationary distribution and for the expected hitting times between any two vertices in the tree. We show that our algorithms improve classical procedures: they do not exhibit ill-posedness and the orders of their complexities are smaller than those of traditional algorithms found in the literature.