Before revision: The main purpose of this work is to prove a theorem about the probability that a random mapping  possesses a unique highest tree in its underlying graph. However, we hope that some of auxiliary statements that we present here can be useful for proving results appealing to the theory of critical Galton-Watson branching processes.
After revision: The main purpose of this work is to prove a theorem about the probability that a random mapping 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 in the proof of the author's result about the probability of being synchronizable for a random automaton. Furthermore, some of auxiliary statements that we present here can be useful for proving results appealing to the theory of critical Galton-Watson branching processes.