  Arthur Cayley famously proved that there are n to the power n-2 labeled trees
on n vertices. Here we go much further and show how to enumerate, fully
automatically, labeled trees such that every vertex has a number of neighbors
that belongs to a specified finite set, and also count trees where the number
of neighbors is not allowed to be in a given finite set. We also give detailed
statistical analysis, and show that in the sample space of labeled trees with n
vertices, the random variable "number of vertices with d neighbors" is
asymptotically normal, and for any different degrees, are jointly
asymptotically normal, but of course, not independently so (except for the pair
(1,3), i.e. the number of leaves and the number of degree-3 vertices, where
there are asymptotically independent). We also give new proofs to Amram Meir
and John Noon's expressions for the limiting expectation and variance for
these, and derive an explicit expression for the covariance.
