Random Hoppe trees

The random Hoppe 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. We analyze the height, internal path length and number of leaves of the Hoppe treewith n nodesaswellasthedepthofthelastinsertednodeasymptotically as n → ∞. Mainly expectations, variances and asymptotic distributions of these parameters are derived.