  An algorithm is designed which decomposes a tropical univariate rational
function into a composition of tropical binomials and trinomials. When a
function is monotone, the composition consists just of binomials. Similar
algorithms are designed for decomposing tropical algebraic rational functions
being (in the classical language) piece-wise linear functions with rational
slopes of their linear pieces. In addition, we provide a criterion when the
composition of two tropical polynomials commutes (for classical polynomials a
similar question was answered by J.~Ritt).
