## 1. Introduction

A tree is a connected acyclic graph with a finite vertex set and finite edge set . The distance between two vertices is the number of edges in the (unique) path in that joins and . In this paper, we focus on counting the pairs of vertices that are some given distance apart, or equivalently the paths of a given length, in a perfect tree.

Given a tree , let denote the number of pairs of vertices at distance exactly from each other. That is,

Note immediately that . Furthermore, from the observations that each vertex of degree is the central vertex of distinct paths of length and that each edge is the central edge of distinct paths of length , we obtain

Similar expressions for when become increasingly complicated.

Faudree et al. [3] constructed examples showing that two non-isomorphic trees can have identical path length distributions (that is, for all ). Tight upper and lower bounds for were given by Dankelmann [1] in terms of and either the radius or diameter of .

A binary tree , in which every vertex has degree or degree , is perfect (or balanced) if has the maximum number of vertices among all binary trees of the same diameter. De Jong et al. [2] used a recursive approach to show that the perfect binary tree of diameter with degree- vertices has

paths of length for .

We adopt a different approach to extend this to perfect -ary trees, where each vertex has degree or . In particular, we prove the following theorem.

###### Theorem 1.1.

Let be the perfect unrooted -ary tree of diameter . Then, for ,

where is the number of vertices in the perfect unrooted -ary tree of diameter .

We first derive an analogous theorem for perfect rooted -ary trees, where the root has degree and all other vertices have degree or . Theorem 1.2 is obtained in Section 2 by counting the paths of length in a perfect rooted

-ary tree according to minimum depth, considering odd

and even separately.###### Theorem 1.2.

Let be the perfect rooted -ary tree of depth , and let satisfy . If is odd, then

and if is even, then

where is the number of vertices in the perfect rooted -ary tree of depth .

## 2. Perfect rooted -ary trees

In a rooted -ary tree , there is a distinguished vertex of degree called the root, while every other vertex has degree or . The depth of is the maximum value of over all vertices . We call perfect if and only if every degree- vertex is distance from the root .

Let be the perfect rooted -ary tree of depth . For , there are exactly vertices for which . Let be a path of length in . Then there is a unique vertex , , such that for all . We call a type- path rooted at .

###### Lemma 2.1.

Let be the perfect rooted -ary tree of depth . If , then the number of type- paths in rooted at is . If , then the number of type- paths in rooted at is

###### Proof.

The case is obvious, as is the case with . Assume that and that . Then any type- path can be decomposed into a type- path rooted at and a type- path rooted at , where these two paths are disjoint. There are choices for the type- path. Once this choice has been made, there are choices for the type- path, so the total number of type- paths rooted at is . If , then this argument counts each type- path twice, hence the third equality. ∎

Let denote the number of paths of length in the perfect rooted -ary tree of depth . The preceding lemma can be used to derive exact expressions for . We consider paths of odd length and paths of even length separately, and make repeated use of the identity

###### Proposition 2.2.

The number of paths of length in the perfect rooted -ary tree of depth , where , is

###### Proof.

Let be the perfect rooted -ary tree of depth . If , then the longest path in has length , and so .

If and , then by Lemma 2.1, there are type- paths rooted at each vertex for which . Therefore,

If , then there are type- paths rooted at each vertex for which . Furthermore, for , there are type- paths rooted at each vertex for which . Therefore,

∎

###### Proposition 2.3.

The number of paths of length in the perfect rooted -ary tree of depth , where , is

###### Proof.

Let be the perfect rooted -ary tree of depth . If , then the longest path in has length , and so .

If , then by Lemma 2.1, there are type- paths rooted at each vertex for which . Furthermore, for , there are type- paths rooted at each vertex for which . Therefore,

If , then by Lemma 2.1, there are type- paths rooted at each vertex for which and type- paths rooted at each vertex for which . Furthermore, for , there are type- paths rooted at each vertex for which . Therefore,

∎

## 3. Perfect unrooted -ary trees

In an unrooted -ary tree , every vertex has degree or . The diameter of is the maximum value of over all pairs of vertices . We call perfect if and only if every degree- vertex is distance from some other vertex.

The symmetry of a perfect unrooted -ary tree of diameter depends on whether is odd or even. We introduce some notation to be used in this respect. If is odd, then can be constructed by connecting the roots of two perfect rooted -ary trees of depth with an edge . If is even, then can be constructed by connecting the roots of the perfect rooted -ary tree of depth and the perfect rooted -ary tree of depth with an edge . In either case, a path in is either contained in , contained in , or contains .

Let denote the number of paths of length in the perfect unrooted -ary tree of diameter . We consider four cases, depending on the parities of and . The proofs of the propositions below make repeated use of Lemma 2.1 and Propositions 2.2 and 2.3.

###### Proposition 3.1.

The number of paths of length in the perfect unrooted -ary tree of diameter , where is

###### Proof.

Let be the perfect unrooted -ary tree of diameter . We use the decomposition of into , , and . The number of paths in of length that contain is

If , then . If , then the depth of (and of ) is , so . Also, , and hence . If , then , and so

If , then , and so

If , then , and so

∎

###### Proposition 3.2.

The number of paths of length in the perfect unrooted -ary tree of diameter , where , is

###### Proof.

Using the decomposition of into , , and , the number of paths of length in that contain is

and again the result is immediate for . If , then , and so

If , then , and so

∎

###### Proposition 3.3.

The number of paths of length in the perfect unrooted -ary tree of diameter , where , is

###### Proof.

Using the decomposition of into (depth ), (depth ), and , the number of paths of length paths in that contain is

If , then . If , then , and so

If , then , and so

If , then , and so

If , then , and so

∎

###### Proposition 3.4.

The number of paths of length in the perfect unrooted -ary tree of diameter , where , is

###### Proof.

Using the decomposition of into (depth ), (depth ), and , the number of paths of length in that contain is

and again the result is immediate for . If , then , and so

If , then , and so

If , then , and so

If , then , and so

∎

## References

- [1] Dankelmann, P. (2011) On the distance distribution of trees. Discrete Appl. Math., 159, pp. 945–952.
- [2] De Jong, J. V., MacLeod, J. C., and Steel, M. (2015) Neighborhoods of Phylogenetic Trees: Exact and Asymptotic Counts. SIAM J. Discr

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