Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $k$-protected vertices in unlabeled rooted plane trees   We find a simple, closed formula for the proportion of vertices which are
$k$-protected in all unlabeled rooted plane trees on $n$ vertices. We also find
that, as $n$ goes to infinity, the average rank of a random vertex in a tree of
size $n$ approaches 0.727649, and the average rank of the root of a tree of
size $n$ approaches 1.62297.

11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
22
The categories of this paper are: chao-dyn
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The categories of this paper are: comp-gas
20
The categories of this paper are: acc-phys
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci