Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Random increasing plane trees: asymptotic enumeration of vertices by
  distance from leaves   We prove that for any fixed $k$, the probability that a random vertex of a
random increasing plane tree is of rank $k$, i.e. at vertex-distance $k$ from
the leaves, converges to a constant $c_k$ as the size $n$ of the tree goes to
infinity. We prove a similar result for randomly selected $r$-tuples of
vertices that shows that the ranks of a finite uniformly random set of vertices
are asymptotically independent. We compute the exact value of $c_k$ for $0\leq
k\leq 3$, demonstrating that the limiting expected fraction of vertices with
rank $\le 3$ is $0.9997\dots$. In general, the distribution $\{c_k\}$ is
quasi-Poisson, i.e. $c_k=O(1/(k-2)!)$. We prove that with high probability, the
highest rank of a vertex in the tree is sandwiched between $\log n / (\log(\log
n))$ and $1.5\log n/(\log(\log n))$.

1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
34
The categories of this paper are: chem-ph