  We prove the exact asymptotic
$1-\left({\frac{2\pi}{3}-\frac{827}{288\pi}}+o(1)\right)/{\sqrt{n}}$ for the
probability that the underlying graph of a random mapping of $n$ elements
possesses a unique highest tree. The property of having a unique highest tree
turned out to be crucial in the solution of the famous Road Coloring Problem as
well as the generalization of this property in the proof of the author's result
about the probability of being synchronizable for a random automaton.
