Large Trees in a Random Mapping Pattern

Mapping patterns may be represented by unlabelled directed graphs in which each point has outdegree one. We consider the uniform probability measure on the set of all mapping patterns on n points and derive the limiting distribution of the size of the largest tree as n → ∞. It is also shown that the asymptotic results concerning the structure of trees of a random mapping pattern coincide with those obtained earlier for the graphs of mappings on labelled point-sets.