Large deviations for the number of trees of a given size and for the maximum size of a tree in a random forest

We consider the set of all forests consisting of $N$ rooted trees such that the roots (and the corresponding trees) are labelled by the numbers $1,\dots,N$, and the remaining $n$ vertices of the forest are labelled by the numbers $1,\dots,n$. Under the assumption that the uniform distribution is defined on this set and $n,N\to\infty$, we prove local limit theorems for the distributions of the random variables equal to the number of trees of a given size and the maximum size of a tree, which permit to estimate the corresponding local probabilities with accuracy of known order, including the probability of large deviations.