Geodesic
On conditions for emergence of a~giant tree in a~random unlabelled forest
Diskretnaya Matematika, Tome 19 (2007) no. 3, pp. 35-50

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We consider the set of random forests consisting of $N$ rooted trees ordered in one of $N!$ possible ways and of $n$ nonroot unlabelled vertices. As $N,n\to\infty$, we find the limit distributions of the $(N-p)$th term of the set of order statistics obtained by arranging the sizes of the trees of a random unlabelled forest in nondescending order for fixed $p=1,2,\dots$ . We find that a giant tree (that is, a tree of size $n+o(n)$) emerges in the only case where $N,n\to\infty$ so that $N/\sqrt n\to0$. 
@article{DM_2007_19_3_a3,
     author = {E. V. Khvorostyanskaya},
     title = {On conditions for emergence of a~giant tree in a~random unlabelled forest},
     journal = {Diskretnaya Matematika},
     pages = {35--50},
     publisher = {mathdoc},
     volume = {19},
     number = {3},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2007_19_3_a3/}
}
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E. V. Khvorostyanskaya. On conditions for emergence of a~giant tree in a~random unlabelled forest. Diskretnaya Matematika, Tome 19 (2007) no. 3, pp. 35-50. http://geodesic.mathdoc.fr/item/DM_2007_19_3_a3/