Locally Oriented Noncrossing Trees

Isaac Owino Okoth,Stephan Wagner

doi:10.37236/5164

Abstract

We define an orientation on the edges of a noncrossing tree induced by the labels: for a noncrossing tree (i.e., the edges do not cross) with vertices $1,2,\ldots,n$ arranged on a circle in this order, all edges are oriented towards the vertex whose label is higher. The main purpose of this paper is to study the distribution of noncrossing trees with respect to the indegree and outdegree sequence determined by this orientation. In particular, an explicit formula for the number of noncrossing trees with given indegree and outdegree sequence is proved and several corollaries are deduced from it. Sources (vertices of indegree $0$) and sinks (vertices of outdegree $0$) play a special role in this context. In particular, it turns out that noncrossing trees with a given number of sources and sinks correspond bijectively to ternary trees with a given number of middle- and right-edges, and an explicit bijection is provided for this fact. Finally, the in- and outdegree distribution of a single vertex is considered and explicit counting formulas are provided again.

Highlights

A classical refinement of Cayley’s formula for the number of labelled trees with n vertices is obtained by taking the vertex degrees into account as well
This gives rise to an outdegree sequence 0e01e12e2 · · ·, where ei is the number of vertices of outdegree i
Du and Yin [4] as well as Shin and Zeng [15] by another method proved that this expression counts the number of labelled trees on n vertices whose outdegree sequence is λ = 0e01e12e2 · · · with respect to a local orientation of the vertices, where every edge is oriented towards the end vertex whose label is higher

Summary

Introduction

A classical refinement of Cayley’s formula for the number of labelled trees with n vertices is obtained by taking the vertex degrees into account as well. Du and Yin [4] as well as Shin and Zeng [15] by another method proved that this expression counts the number of labelled trees on n vertices whose outdegree sequence is λ = 0e01e12e2 · · · with respect to a local orientation of the vertices (as opposed to the global orientation away from the root), where every edge is oriented towards the end vertex whose label is higher. In [6], Flajolet and Noy obtained a formula for the number of noncrossing trees on n vertices with a given outdegree sequence with respect to the aforementioned global orientation, where all edges are oriented to point away from the root. If we consider a random tree of order n with a given number e0 of sources (vertices of indegree 0) and a given number d0 of sinks (vertices of outdegree 0), the in- and outdegree sequence are independent random variables

Proof of the main theorem

Consequences of the main theorem

A bijection with ternary trees

In- and outdegree of a single vertex