2-Binary trees: Bijections and related issues

Bijections between certain families of labelled and unlabelled d-ary trees

“…In [2,6,7] remarkable enumeration formulae and relations between subclasses of two-coloured trees, where restrictions on the colours of two connected nodes are made, and other combinatorial objects are obtained. In particular in [2,7] it has been shown in a bijective way that the number of -free two-coloured binary trees with a black root and of size n ≥ 1 are equal to the number of (uncoloured) ternary trees of size n. Here a two-coloured binary tree is called -free, when there is no edge occurring in the tree that connects a parent coloured black with a right child coloured white. The bijection presented in [7] gives a procedure, which is easy to implement and that allows to encode (and decode) a ternary tree by a -free two-coloured binary tree with a black root of the same size.…”

“…, the d-th child, where we have to allow "empty children" ε. This recursive description can also be expressed via the following formal equation for T d , where∪ denotes the disjoint union of two combinatorial families: (1) T d = ε∪ It is well known and can be shown in many ways that the number T (d) n of d-ary trees of size n, where the size |T | of a tree T is here always measured by its number of nodes, is given by the generalized Catalan numbers: (2) T (d) n = 1 (d − 1)n + 1 dn n , for n ≥ 0.…”

A bijection between a set of lexicographic semiorders and pairs of non-crossing Dyck paths

“…Similar to semiorders, it is well-known that the number of distinct Dyck paths of length 2n is the nth Catalan number (e.g., Stanley, 1999), see also Deutsch (1999). Stanley (1999) gives an overview of the bijective relationships between Dyck paths and many other mathematical structures, including binary trees (see also Gu, Li, & Mansour, 2008), triangulations of an (n+2)-gon, ways to parenthesize a string of length n+1, and ballot sequences. Balof and Menashe (2007) present a bijection …”

Consecutive Pattern Avoidances in Non-crossing Trees

“…A non-crossing tree (NC-tree for short) is a tree drawn on n points in {1, 2, · · · , n} numbered in counterclockwise order on a circle such that the edges lie entirely within the circle and do not cross. Non-crossing trees have been investigated by Chen and Yan [1], Deutsch and Noy [3], Flajolet and Noy [4], Gu, et al [5], Hough [6], Noy [8], Panholzer and Prodinger [9]. Recently, some problems of pattern avoidance in NC-trees have been studied by Sun and Wang [13].…”