Báo cáo toán học: "A simple bijection between binary trees and colored ternary trees"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: A simple bijection between binary trees and colored ternary trees. | A simple bijection between binary trees and colored ternary trees Yidong Sun Department of Mathematics Dalian Maritime University 116026 Dalian . China sydmath@ Submitted Feb 25 2009 Accepted Mar 28 2010 Published Apr 5 2010 Mathematics Subject Classification 05C05 05A19 Abstract In this short note we first present a simple bijection between binary trees and colored ternary trees and then derive a new identity related to generalized Catalan numbers. Keywords Binary tree Ternary tree Generalized Catalan number. 1 Introduction Recently Mansour and the author 2 obtained an identity involving 2-Catalan numbers Cn2 TC-W 2n l and 3-Catalan numbers C n 2 2n l k nJ tA A 1 . 3n l n y2 1 3p 1Wn p A 3P A p A 3P 1 f2n 1 2n 1 n J 1-1 In this short note we first present a simple bijection between complete binary trees and colored complete ternary trees and then derive the following generalized identity n 2 m i3p m in p m 1 3p m p J n 2p m i2n m 2n m n 2 A bijective algorithm for binary and ternary trees A colored ternary trees is a complete ternary tree such that all its vertices are signed a nonnegative integer called color number. Let Tn p denote the set of colored ternary trees THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N20 1 T with p internal vertices such that the sum of all the color numbers of T is n 2p. Define Tn U Tn . Let Bn denote the set of complete binary trees with n internal vertices. For any B G Bn let P V1V2 Vk be a path of length k of B viewed from the root of B . P is called a R-path if 1 Vi is the right child of Vi-1 for 2 i k and 2 the left child of Vi is a leaf for 1 i k. In addition P is called a maximal R-path if there exists no vertex u such that uP or Pu forms a R-path. P is called an L-path if k 2 and Vi is the left child of vi-1 for 2 i k. P is called a maximal L-path if there exists no vertex u such that uP or Pu forms an L-path. Clearly a leaf can never be R-path or L-path. Note that the definition of L-path is different .