Decorating trees grown in urns

Abstract: Random recursive trees are classic models of random trees. A random recursive tree is initiated with a single root vertex and constructed in steps, whereby at each step a vertex is added as the child of a vertex chosen uniformly at random in the tree. A preferential attachment tree is constructed in a similar manner, except the random choice of the vertex at each step is made proportional to its outdegree.The models studied in this thesis are generalizations of random recursive trees and preferential attachment trees. Hooking networks are constructed recursively, whereby a new graph called a block is attached at each step, instead of a single vertex. Bipolar networks are directed graphs recursively constructed by choosing an arc at random in the network and replacing it with a directed graph. Random recursive metric spaces are similar to hooking networks, but the blocks attached at each step are metric spaces. Finally, a broadcast induced colouring on a random recursive tree or preferential attachment tree is a random 2-colouring of the vertices as red or blue in the following way. The root vertex is coloured red or blue with equal probability, and every other vertex takes the colour of its parent with probability p and the other colour with probability 1-p.In Paper I, we prove normal limit laws for the degree distributions of hooking networks and bipolar networks. Paper II provides a normal limit law, under certain conditions, for the insertion depth in hooking networks; the distance from the initial starting block to the newly added block. In Paper III, a similar normal limit law is proved for the insertion depth in random recursive metric spaces. Broadcast induced colourings in random recursive trees and preferential attachment trees are studied in Paper IV, where we prove limit laws for the number of vertices of each colour, the number of clusters (maximal monochromatic subtrees) of each colour, as well as the number of leaves of each colour and the number of 2-coloured trees appearing in the fringe. We also prove limit laws for the size of the cluster containing the root vertex.