Counting Contours on Trees

TL;DR

This paper derives exact counts and estimates for contours in d-ary and general trees, and characterizes trees with infinitely many contours of the same size containing a fixed vertex.

Contribution

It provides the first exact enumeration of contours in d-ary trees and characterizes trees with infinitely many such contours.

Findings

Exact number of contours in d-ary trees calculated

Sharp estimates for contours in general trees provided

Characterization of trees with infinitely many contours of the same size

Abstract

We calculate the exact number of contours of size $n$ containing a fixed vertex in $d$-ary trees and provide sharp estimates for this number for more general trees. We also obtain a characterization of the locally finite trees with infinitely many contours of the same size containing a fixed vertex.