Before revision: 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.
After revision: We prove the exact asymptotic 1- \Theta(\frac{1 \left({{3}-827{288\pi}}}+o(1)\right) for the probability that the underlying graph of a random mapping of n elements possesses a unique highest tree.