A Bijection on Ordered Trees and Its Consequences

Abstract

A bijection is introduced in the set of all ordered trees having n edges from which one derives that, for each positive integer q, the parameters "number of nodes of degree q" and "number of odd-level nodes of degree q-1" are equidistributed.

A New Bijection Between RNA Secondary Structures and Plane Trees and its Consequences

“…Figure 2 gives an example of the bijection ϕ. The following corollary which is implied in [4] can be obtained as well. Proof.…”

“…Therefore, the Schmitt-Waterman bijection maps an RNA secondary structure with k isolated bases to a plane tree with k leaves. Thus, combined with our bijection φ, with RNA secondary structures serving as intermediate objects, we immediately obtain the following well-known result [4].…”

“…A dual perspective which appears to be ignored (at least less-studied) is that, vertically, a plane tree can be decomposed into paths associated to its leaves. Through our new bijection ϕ on plane trees, not only can we show the correspondence between horizontal fibers and odd-level vertices established by Deutsch [4], but we can also show that the vertical paths associated to leaves actually correspond to even-level vertices at the same time. Namely, we discover that the joint distribution of horizontal fibers and vertical paths is the same as the joint distribution of odd-and even-level vertices.…”

“…The most relevant studies on plane trees in the literature are as follows. In his paper [4], Deutsch presented an implicit, iteratively constructed bijection on plane trees which allowed him to show that the number of vertices of degree q > 0 and the number of odd-level vertices of degree q − 1 are equidistributed on the set of all plane trees. In particular, the case q = 1 implies that the number of plane trees with k leaves is the same as the number of plane trees with k even-level vertices, where the former is well known to be the Narayana number.…”

A Bijection on Dyck Paths and its Cycle Structure

“…1 Introduction There are several bijections on Dyck paths in the literature [1,2,3,4,5,6,7,8,9,10,11], usually introduced to show the equidistribution of statistics: if a bijection sends statistic A to statistic B, then clearly both have the same distribution. Another aspect of such a bijection is its cycle structure considered as a permutation on Dyck paths.…”