Limit theorems for the number of trees of a given size in a random forest

The author considers the set of all forests consisting of $N$ rooted trees and containing $n$ nonroot vertices; the root vertices are numbered from 1 to $N$, and the nonroot from 1 to $n$. A uniform probability distribution is introduced on this set. Let $\mu_r(n,N)$ denote a random variable equal to the number of trees of a random forest containing exactly $r$ nonroot vertices. Results are obtained yielding a complete description of the limit behavior of the variables $\mu_r(n,N)$ for all values of $r$ for various ways of letting $n$ and $N$ approach infinity. It is shown that these results can be used for studying random mappings.
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