  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.
