We prove the exact asymptotic 1- \Theta(\frac{1 \left({{3}-827{288\pi}}}\right)  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 \mbox{%DIFAUXCMD TRRCP08 as well as  in the proof of the author's result about the probability of being synchronizable for a random automaton \mbox{%DIFAUXCMD RandSynch .
We prove the exact asymptotic 1- \left({\frac{2\pi{3}-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 .
D
\Theta(\frac{1
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\frac{2\pi
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+o(1)
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/
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\sqrt{n
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\mbox{%DIFAUXCMD TRRCP08
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the generalization of this property
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\mbox{%DIFAUXCMD RandSynch