Geodesic
Large deviations for the number of trees of a~given size and for the maximum size of a~tree in a~random forest
Diskretnaya Matematika, Tome 18 (2006) no. 3, pp. 77-84.

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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. 
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A. N. Timashev. Large deviations for the number of trees of a~given size and for the maximum size of a~tree in a~random forest. Diskretnaya Matematika, Tome 18 (2006) no. 3, pp. 77-84. http://geodesic.mathdoc.fr/item/DM_2006_18_3_a4/

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