  In this paper we show that given a max-plus automaton (over trees, and with
real weights) computing a function $f$ and a min-plus automaton (similar)
computing a function $g$ such that $f\leqslant g$, there exists effectively an
unambiguous tropical automaton computing $h$ such that $f\leqslant h\leqslant
g$. This generalizes a result of Lombardy and Mairesse of 2006 stating that
series which are both max-plus and min-plus rational are unambiguous. This
generalization goes in two directions: trees are considered instead of words,
and separation is established instead of characterization (separation implies
characterization). The techniques in the two proofs are very different.
