Symmetry in Randomness : Additive Functionals and Symmetries of Random Trees and Tree-Like Graphs

Properties of symmetries in random trees and tree-like graphs are explored. The primary structures studied are Galton-Watson trees, unlabeled unordered trees as well as labeled subcritical graphs. The most significant results are exponential decay of the probability that two trees are isomorphic for some types of Galton-Watson trees and a central limit theorem for the logarithm of the size of the automorphism in all of the three models listed above, but a number of related theorems are also given including the limiting distribution of the number of labelings of an unlabeled unordered tree. An important tool is that of additive functionals of rooted trees and we also show how to extend the definition to the case of subcritical graphs together with powerful results that were previously only known for trees. Methods from both probability theory and analytic combinatorics are used.