Fringe analysis of plane trees related to cutting and pruning


Rooted plane trees are reduced by four different operations on the fringe. The number of surviving nodes after reducing the tree repeatedly for a fixed number of times is asymptotically analyzed. The four different operations include cutting all or only the leftmost leaves or maximal paths. This generalizes the concept of pruning a tree. The results include exact expressions and asymptotic expansions for the expected value and the variance as well as central limit theorems.


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Corresponding author

Correspondence to Benjamin Hackl.

Additional information

Benjamin Hackl and Clemens Heuberger are supported by the Austrian Science Fund (FWF): P 24644-N26 and by the Karl Popper Kolleg “Modeling-Simulation-Optimization” funded by the Alpen-Adria-Universität Klagenfurt and by the Carinthian Economic Promotion Fund (KWF). Helmut Prodinger is supported by an incentive grant of the National Research Foundation of South Africa. Part of this author’s work was done while he visited Academia Sinica. He thanks the Institute of Statistical Science for its hospitality.

This is the full version of the extended abstract [12].

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Hackl, B., Heuberger, C., Kropf, S. et al. Fringe analysis of plane trees related to cutting and pruning. Aequat. Math. 92, 311–353 (2018).

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Mathematics Subject Classification

  • 05A16
  • 05C05
  • 05A15
  • 05A19
  • 60C05


  • Plane trees
  • Pruning
  • Tree reductions
  • Central limit theorem
  • Narayana polynomials