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Arthur Cayley famously proved that there are n to the power n-2 labeled treeson n vertices. Here we go much further and show how to enumerate, fullyautomatically, labeled trees such that every vertex has a number of neighborsthat belongs to a specified finite set, and also count trees where the numberof neighbors is not allowed to be in a given finite set. We also give detailedstatistical analysis, and show that in the sample space of labeled trees with nvertices, the random variable "number of vertices with d neighbors" isasymptotically normal, and for any different degrees, are jointlyasymptotically 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, wherethere are asymptotically independent). We also give new proofs to Amram Meirand John Noon's expressions for the limiting expectation and variance forthese, and derive an explicit expression for the covariance.

Automated Counting and Statistical Analysis of Labeled Trees with Degree Restrictions