Highest Trees of Random Mappings

We prove the exact asymptotic 1 − Θ( 1 √ 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 [7] as well as in the proof of the author’s result about the probability of being synchronizable for a random automaton [1]. Furthermore, some of auxiliary statements that we present here can be useful for solving problems appearing in the theory of critical Galton-Watson branching processes. 1998 ACM Subject Classification G.2.2