Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counting trees using symmetries   We prove a new formula for the generating function of multitype Cayley trees
counted according to their degree distribution. Using this formula we recover
and extend several enumerative results about trees. In particular, we extend
some results by Knuth and by Bousquet-M\'elou and Chapuy about embedded trees.
We also give a new proof of the multivariate Lagrange inversion formula. Our
strategy for counting trees is to exploit symmetries of refined enumeration
formulas: proving these symmetries is easy, and once the symmetries are proved
the formulas follow effortlessly. We also adapt this strategy to recover an
enumeration formula of Goulden and Jackson for cacti counted according to their
degree distribution.

17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin
18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
14
The categories of this paper are: math-ph
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph