Reducing Simply Generated Trees by Iterative Leaf Cutting

We consider a procedure to reduce simply generated trees by iteratively removing all leaves. In the context of this reduction, we study the number of vertices that are deleted after applying this procedure a fixed number of times by using an additive tree parameter model combined with a recursive characterization. Our results include asymptotic formulas for mean and variance of this quantity as well as a central limit theorem.

[1]  Ronald L. Graham,et al.  Concrete mathematics - a foundation for computer science , 1991 .

[2]  Dimbinaina Ralaivaosaona,et al.  Additive functionals of $d$-ary increasing trees , 2016 .

[3]  Higher dimensional quasi-power theorem and Berry–Esseen inequality , 2016, Monatshefte für Mathematik.

[4]  Svante Janson,et al.  Asymptotic normality of fringe subtrees and additive functionals in conditioned Galton–Watson trees , 2013, Random Struct. Algorithms.

[5]  Hsien-Kuei Hwang,et al.  On Convergence Rates in the Central Limit Theorems for Combinatorial Structures , 1998, Eur. J. Comb..

[6]  Helmut Prodinger,et al.  Reductions of binary trees and lattice paths induced by the register function , 2016, Theor. Comput. Sci..

[7]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[8]  Svante Janson,et al.  Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation , 2011, 1112.0510.

[9]  Philippe Flajolet,et al.  Singularity Analysis of Generating Functions , 1990, SIAM J. Discret. Math..

[10]  David Aldous,et al.  Asymptotic Fringe Distributions for General Families of Random Trees , 1991 .

[11]  Helmut Prodinger,et al.  Growing and Destroying Catalan-Stanley Trees , 2017, Discret. Math. Theor. Comput. Sci..

[12]  STEPHAN WAGNER,et al.  Central Limit Theorems for Additive Tree Parameters with Small Toll Functions , 2014, Combinatorics, Probability and Computing.

[13]  Helmut Prodinger,et al.  Fringe analysis of plane trees related to cutting and pruning , 2017, Aequationes mathematicae.