Journal of Applied Probability

Asymptotic analysis of Hoppe trees

Kevin Leckey and Ralph Neininger

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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.

Article information

J. Appl. Probab., Volume 50, Number 1 (2013), 228-238.

First available in Project Euclid: 20 March 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability
Secondary: 60G42: Martingales with discrete parameter 68R05: Combinatorics

Hoppe urn random tree weak convergence martingale combinatorial probability


Leckey, Kevin; Neininger, Ralph. Asymptotic analysis of Hoppe trees. J. Appl. Probab. 50 (2013), no. 1, 228--238. doi:10.1239/jap/1363784435.

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