Asymptotic Analysis of Hoppe Trees

We introduce and analyze a random tree model associated to Hoppe’s urn. The tree is built successively by adding nodes to the existing tree when starting with the single root node. In each step a node is added to the tree as a child of an existing node where these parent nodes are chosen randomly with probabilities proportional to their weights. The root node has weight θ > 0, a given fixed parameter, all other nodes have weight 1. This resembles the stochastic dynamic of Hoppe’s urn. For θ = 1 the resulting tree is the well-studied random recursive tree. We analyze the height, internal path length and number of leaves of the Hoppe tree with n nodes as well as the depth of the last inserted node asymptotically as n → ∞. Mainly expectations, variances and asymptotic distributions of these parameters are derived. AMS 2010 subject classifications. Primary 60F05, 60C05; secondary 60G42, 68R05.