The Cycle Lemma and Some Applications

Two proofs of a frequently rediscovered combinatorial lemma are presented. Using the lemma, a combinatorial proof is given that the average height of an ordered (plane-planted) tree is approximately twice the average node (vertex) level.

[1]  J. Moon,et al.  On the Altitude of Nodes in Random Trees , 1978, Canadian Journal of Mathematics.

[2]  Ira M. Gessel,et al.  A combinatorial proof of the multivariable lagrange inversion formula , 1987, J. Comb. Theory, Ser. A.

[3]  S. G. Mohanty,et al.  Lattice Path Counting and Applications. , 1980 .

[4]  Nachum Dershowitz,et al.  Applied Tree Enumerations , 1981, CAAP.

[5]  George M. Bergman.  Terms and cyclic permutations , 1978 .

[6]  Lajos Takács,et al.  Combinatorial Methods in the Theory of Stochastic Processes , 1967 .

[7]  A. D. Sands,et al.  On generalised catalan numbers , 1978, Discret. Math..

[8]  Philippe Flajolet,et al.  The Average Height of Binary Trees and Other Simple Trees , 1982, J. Comput. Syst. Sci..

[9]  Peter Hall.  Some Word‐Problems , 1958 .

[10]  David Singmaster,et al.  An Elementary Evaluation of the Catalan Numbers , 1978 .

[11]  T. Motzkin,et al.  Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products , 1948 .

[12]  C. L. Mallows,et al.  Some Aspects of the Random Sequence , 1965 .

[13]  Donald E. Knuth.  The art of computer programming: fundamental algorithms , 1969 .

[14]  F. Spitzer.  A Combinatorial Lemma and its Application to Probability Theory , 1956 .

[15]  de Ng Dick Bruijn,et al.  THE AVERAGE HEIGHT OF PLANTED PLANE TREES , 1972 .

[16]  D. Klarner,et al.  Correspondences between plane trees and binary sequences , 1970 .

[17]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[18]  Cheng Yang,et al.  A Transformation on Ordered Trees , 1980, Comput. J..

[19]  G. N. Raney.  Functional composition patterns and power series reversion , 1960 .

[20]  T. Motzkin,et al.  A problem of arrangements , 1947 .

[21]  Arthur Cayley,et al.  The Collected Mathematical Papers: On the analytical forms called trees , 1881 .

[22]  Nachum Dershowitz,et al.  Patterns in trees , 1989, Discret. Appl. Math..