On one characteristic of a random mapping with given number of cycles

We consider the random variable equal to the number of trees of given size in the graph of a random one-to-one mapping of an $n$-element set into itself with $m$ connected components. We obtain limit theorems which describe the distribution of this characteristic in the case where $n\to\infty$, $m/\ln n\to\infty$, $m/\ln n=O(\ln n)$.