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\begin{center}
\vskip 1cm{\LARGE\bf
On the Average Path Length of Complete\\
\vskip .12in
$m$-ary Trees
}
\vskip 1cm
\large
M. A. Nyblom \\
School of Mathematics and Geospatial Science\\
RMIT University\\
Melbourne, Victoria 3001\\
Australia\\
\href{mailto:michael.nyblom@rmit.edu.au}{\tt michael.nyblom@rmit.edu.au} \\
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\begin{abstract}
Define the average path length in a connected graph as the sum of the
length of the shortest path between all pairs of nodes, divided by the
total number of pairs of nodes. Letting $S_N$ denote the sum of the
shortest path lengths between all pairs of nodes in a complete $m$-ary
tree of depth $N$, we derive a first-order linear but non-homogeneous
recurrence relation for $S_{N}$, from which a closed-form expression
for $S_{N}$ is obtained. Using this explicit expression for $S_{N}$, we
show that the average path length within this graph/network is
asymptotic to $D-\frac{4}{m-1}$, where $D$ is the diameter of the
$m$-ary tree, that is, the longest shortest path.  This asymptotic
estimate for the average path length confirms a conjectured asymptotic
estimate in the case of complete binary tree.
\end{abstract}


\section{Introduction}

A network is a graph $G=(E,V)$ having no loops, which is also connected in that one can find a sequence of edges in $E$, or path, connecting any two pairs of vertices in $V$. The study of networks has in recent times blossomed into a new field which has found applications in such diverse areas as communication engineering and sociology. One focus of attention in this new discipline area, coined ``Network Science" \cite{lew}, is the problem of designing networks in an economic way, whereby a sequence of edges a path within the network can be constructed to connect any two pairs of nodes, while keeping the total number of edges used to a minimum. Such highly edge-efficient networks are useful in the area of designing communication networks, as they help reduce transmission delays. One way to compare the edge-efficiency of different networks $G=(E,V)$, is to introduce a new metric \cite[p.\ 74]{lew}, known as the edge efficiency of $G$, and
defined by
\[
E(G)=1-\frac{\bar{P}(G)}{|E|}\mbox{ ,}
\]
where $\bar{P}(G)$ is the average path length of $G$, which is calculated as the sum of the length of the shortest paths connecting all pairs of nodes in $V$, divided by the total number of pairs of nodes in $V$. 
(Note if there is more than one shortest path connecting a pair of nodes, then the length of these paths will still occur once as a summand used in the calculation of $\bar{P}(G)$.)

One can view $E(G)$ as a measure of how effectively edges are used within a network to minimize the average path length. As $\bar{P}(G)\leq |E|$ we see that $E(G)$ ranges from $0$ when $\bar{P}(G)=|E|$,
to approximately $1$ when $\bar{P}(G)\cong 0$. The case $E(G)=0$ corresponds to a worst case when the average path length is as large as possible, while if $E(G)\cong 1$ then the average path length is small relative to the number of edges, and so every edge can be seen as contributing to the edge efficiency of the network. A network in which $E(G)$ approaches 1 as $|E|$ approaches infinity has been described as being ``scalable'' \cite[p.\ 74]{lew}. To experimentally
demostrate this scalable property of such regular networks as the complete binary trees, T. Lewis \cite[p.\ 76]{lew} developed a breadth-first algorithm for the computation of $\bar{P}(G)$, for various increasing depths $N$ of these binary trees. As a consequence of his experimental results, it was conjectured that for a complete binary tree $\bar{P}(G)\sim D-4$, as $N\to\infty$, where $D$ is the diameter of the tree, that is, the longest shortest path corresponding to the value of $N$. It should be noted that the diameter is a frequently used characteristic of trees within the literature \cite{liu}.

In this paper we shall prove this asymptotic estimate by establishing a more general result, namely that $\bar{P}(G)\sim D-\frac{4}{m-1}$ as $N\to\infty$, in the case when $G=(E,V)$ is a complete $m$-ary tree, that is, a rooted tree in which all nodes have exactly $m$ children, with the exception of the leaf nodes that have no children, and which are all located at the highest depth (see Figure \ref{F1}). To achieve this end, we first shall derive a closed-form expression for the sum, denoted $S_{N}$, of the shortest path lengths between all pairs of nodes in a complete $m$-ary tree of depth $N$. As a consequence of this expression in (\ref{eq:1}), it will then be an easy task to show that $\bar{P}(G)=\frac{S_{N}}{{ |V| \choose 2}}\sim D-\frac{4}{m-1}$ as $N\to\infty$. It should be noted that the quantity measured by $S_{N}$, is also referred to as a 
{\it Wiener index} in the area of computational chemistry, where graphs such as rooted trees represent the molecular graphs of certain chemical compounds. Although there is a well-known recursive procedure for the calculation of the Wiener index of such graphs \cite{can}, it has been observed \cite{dob} that this general procedure can be cumbersome to apply for particular families of rooted trees having a regular graph structure, such as the complete $m$-ary trees. It is for this reason we have in this paper directly exploited the graph structure of the $m$-ary tree, to produce a first-order linear but non-homogeneous recurrence relation for the calculation of $S_{N}$.

\section{Main result}
\label{sectwo}
To help establish the main result we begin by making two simple observations concerning the complete $m$-ary tree. With reference to Figure \ref{F1}, if the schematic diagram represents a complete
$m$-ary tree of depth $N+1$ in which the root node is at depth $1$, then note that the $m$ subtrees whose root nodes are located at depth $2$ must also be complete $m$-ary trees of depth $N$. Consequently one should be able to express $S_{N+1}$ in terms of $S_{N}$, so establishing a recurrence relation for the sequence $\{ S_{N}\}$ for $N\geq 2$. Secondly observe that as each node, with the exception of the root node, has a unique parent node located at the previous depth, there can only be one shortest path connecting any pair of nodes in a complete $m$-ary tree. It should be noted that uniqueness of the shortest path is a property enjoyed by all trees.

\begin{figure}[H]
\centering
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;
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\caption{An $m$-ary tree of depth $N+1$} \label{F1}
\end{figure}

We shall also need the following elementary technical lemma.
\begin{lemma}\label{L1}
For any $x\in\mathbb{R}\backslash\{ 1\}$ and integer $N\geq 1$
\[
\sum_{n=1}^{N}nx^{n-1}=\frac{1}{(x-1)^{2}}(Nx^{N+1}-Nx^{N}-x^{N}+1)\mbox{ .}
\]
\end{lemma}
\begin{proof}
Differentiate both sides of the identity $1+x+x^{2}+\cdots x^{N}=(x^{N+1}-1)/(x-1)$ with respect to $x$.
\end{proof}

By exploiting the above simple facts, one can now obtain a closed-form expression for $S_{N}$ as follows.
\begin{proposition}\label{P1}
If $G=(E,V)$ is a complete $m$-ary tree of depth $N\geq 2$, then the sum of the shortest path lengths between all pairs of nodes in $V$ is given by
\begin{equation}
S_{N}=m^{2N}\left( \frac{N}{(m-1)^{2}} -\frac{m+1}{(m-1)^{3}}\right) + m^{N}\left( \frac{N}{(m-1)^{2}} +\frac{m+1}{(m-1)^{3}}\right)\mbox{ .}\label{eq:1}
\end{equation}
\end{proposition}
\begin{proof}
The argument used to establish (\ref{eq:1}) will be broken into two
parts. In Part 1 we use the geometry of the complete $m$-ary tree to
derive a first-order linear but non-homogeneous recurrence relation for
$S_{N}$. Although the process of solving such recurrence relations is
standard a sketch of the details leading towards (\ref{eq:1}) is
outlined in Part 2, for completeness.

\bigskip

\noindent{\bf Part 1: }Again with reference to Figure \ref{F1}, observe that the set $S$ of shortest paths connecting pairs of nodes in a complete $m$-ary tree of depth $N+1$, can be partitioned as
$S={\cal S}_{1}\cup {\cal S}_{2}\cup {\cal S}_{3}$, where ${\cal S}_{1}$ is the set of shortest paths connecting pairs of nodes located in any one but only one of the $m$ subtrees of depth $N$, ${\cal S}_{2}$ is the set of shortest paths connecting any node contained in the $m$ subtrees of depth $N$ to the root of the tree of depth $N+1$, and finally ${\cal S}_{3}$ is the set of shortest paths connecting pairs of nodes located in any two separate subtrees of depth $N$. Clearly by definition, the sum of the lengths of the elements in ${\cal S}_{1}$ must be $mS_{N}$. If we denote the sum of the lengths of the elements in ${\cal S}_{2}$ and ${\cal S}_{3}$ by $X_{N}$ and $Y_{N}$ respectively, we can conclude that
\begin{equation}
S_{N+1}=mS_{N}+X_{N}+Y_{N}\mbox{ ,}\label{eq:2}
\end{equation}
for $N\geq 2$. We now determine explicit expressions for $X_{N}$ and $Y_{N}$, in terms of $m$ and $N$. Considering any one of the $m$ subtrees of depth $N$ in Figure \ref{F1}, recall from definition that the total number of nodes found at a depth of $n$, were $1\leq n\leq N$, is $m^{n-1}$. Now in general from the geometry of a complete $m$-ary tree, observe that the length of the shortest path connecting any node at depth say $i$, to the root of the tree must be $i-1$. As there is only one edge connecting the root of the $m$ subtrees, to the root of the tree of Figure \ref{F1}, we conclude that the length of the shortest path connecting a node at a depth $n$ in any subtree, to the root of the tree is $(n-1)+1=n$. Consequently the sum of the shortest path lengths connecting all nodes in any one subtree of depth $N$, to the root of the tree is $\sum_{n=1}^{N}nm^{n-1}$. Thus by Lemma \ref{L1}, one concludes after setting $x=m$ that
\[
X_{N}=m\sum_{n=1}^{N}nm^{n-1}=\frac{m}{(m-1)^{2}}(Nm^{N+1}-Nm^{N}-m^{N}+1)\mbox{ .}
\]
To determine $Y_{N}$, we first must show that the sum of the lengths of the shortest paths connecting any one node of a given subtree, at a depth $1\leq n\leq N$, to all nodes located in a separate subtree of Figure \ref{F1}, is given by
\begin{equation}
S(n)=(n+1)m^{0}+(n+2)m^{1}+(n+3)m^{2}+\cdots +(n+N)m^{N-1}\mbox{ .}\label{eq:3}
\end{equation}
To this end, recall that the length of the shortest path from a node at depth $1\leq n\leq N$ in any subtree, to the root of the tree is $n$. Consequently a shortest path beginning at a node at depth $1\leq n\leq N$ in one subtree, to a node at depth $1\leq i\leq N$ in another subtree, must first traverse a path of length $n$ from the initial subtree to the root, then a path of length $1$ to the root of the other subtree, and finally a path of length $i-1$ from this root to the node at depth $1\leq i\leq N$ in the second subtree. Thus the length of such a shortest a path is $n+1+i-1=n+i$, and as there are $m^{i-1}$ nodes in the second subtree at depth $i$, we can conclude the sum of these individual shortest path lengths starting at a node at depth $n$ in one subtree, and terminating at the nodes at depth $i$ in another subtree is $(n+i)m^{i-1}$. Finally by adding the terms $(n+i)m^{i-1}$ over all depths $1\leq i\leq N$ in the second subtree, one arrives at the expression in (\ref{eq:3}).

Now as there are $m^{n-1}$ nodes at a depth $n$ in any one subtree of Figure \ref{F1}, observe that $\sum_{n=1}^{N}m^{n-1}S(n)$ must represent the total sum of the shortest path lengths connecting pairs of nodes located in any two separate subtrees of depth $N$. Hence by choosing the $m$ subtrees two at a time, we deduce from definition that $Y_{N}={m \choose 2}\sum_{n=1}^{N}m^{n-1}S(n)$. To determine $Y_{N}$ explicitly first write $S(n)$ as follows
\begin{equation}
S(n)= \sum_{i=1}^{N}(n+i)m^{i-1} = n\sum_{i=1}^{N}m^{i-1} +\sum_{i=1}^{N}im^{i-1} = n\frac{m^{N}-1}{m-1} +\sum_{i=1}^{N}im^{i-1}\mbox{ .}\label{eq:4}
\end{equation}
Then after substituting the right hand side of (\ref{eq:4}) into the above expression for $Y_{N}$, one finds after an application of Lemma \ref{L1} that
\begin{eqnarray*}
Y_{N} &amp; = &amp; {m \choose 2}\sum_{n=1}^{N}m^{n-1}\left( n\left(\frac{m^{N}-1}{m-1}\right) +\sum_{i=1}^{N}im^{i-1}\right) \\
      &amp; = &amp; {m \choose 2}\left( \left( \frac{m^{N}-1}{m-1}\right)\sum_{n=1}^{N}nm^{n-1} +\sum_{n=1}^{N}m^{n-1}\sum_{i=1}^{N}im^{i-1}\right) \\
      &amp; = &amp; {m \choose 2}\left( \left(\frac{m^{N}-1}{m-1}\right)\sum_{n=1}^{N}nm^{n-1} +\left(\frac{m^{N}-1}{m-1}\right)\sum_{i=1}^{N}im^{i-1}\right)\\
      &amp; = &amp; 2{m \choose 2}\left(\frac{m^{N}-1}{m-1}\right)\sum_{n=1}^{N}nm^{n-1}\\
      &amp; = &amp; (m^{N+1}-m)\left(\frac{Nm^{N+1}-Nm^{N}-m^{N}+1}{(m-1)^{2}}\right)\mbox{ .}
\end{eqnarray*}
Adding these explicit expressions for $X_{N}$ and $Y_{N}$ and substituting the result into (\ref{eq:2}), produces the required recurrence relation for $S_{N}$ as follows
\begin{equation}
S_{N+1}-mS_{N}=\left(\frac{m}{m-1}N-\frac{m}{(m-1)^{2}}\right) m^{2N} +\frac{m}{(m-1)^{2}}m^{N}\mbox{ ,}\label{eq:5}
\end{equation}
for $N\geq 2$. The initial condition is clearly given by $S_{2}=m+2{m \choose 2}=m^{2}$.

\bigskip

\noindent{\bf Part 2: }As the recurrence relation in (\ref{eq:5}) is linear, recall that the solution $S_{N}$ must be of the form $S_{N}=Am^{N}+BNm^{N}+(CN+D)m^{2N}$, where $Am^{N}$ corresponds to a general homogeneous solution of (\ref{eq:5}), and the constants $B,C,D$ will be chosen so that the terms $BNm^{N}$ and $(CN+D)m^{2N}$ satisfy (\ref{eq:5}), with a corresponding right-hand side equal to $\frac{m}{(m-1)^{2}}m^{N}$ and $(\frac{m}{m-1}N-\frac{m}{(m-1)^{2}})m^{2N}$ respectively. A standard calculation reveals that $B=C=\frac{1}{(m-1)^{2}}$ and $D=-\frac{m+1}{(m-1)^{3}}$ and so
\[
S_{N}=Am^{N}+\frac{N}{(m-1)^{2}}m^{N}+\left(\frac{N}{(m-1)^{2}}-\frac{m+1}{(m-1)^{3}}\right) m^{2N}\mbox{ .}
\]
Applying the initial condition $S_{2}=m^{2}$ one finally deduces that $A=\frac{m+1}{(m-1)^{3}}$, which results in the closed-form expression in (\ref{eq:1}).
\end{proof}

To state and prove the main result, recall that the diameter $D$ of a network $G=(E,V)$, is the length of the longest shortest path connecting two nodes in $V$. In the case of a complete $m$-ary tree of depth $N\geq 2$, the diameter must be the length of the path connecting the left-most leaf node to the right-most leaf node of the tree, and so $D=2(N-1)$.
\begin{theorem}\label{T1}
For a network $G=(E,V)$ which is a complete $m$-ary tree, the average path length $\bar{P}(G)$ satisfies the following asymptotic estimate
\[
\bar{P}(G)\sim D-\frac{4}{m-1}\quad \mbox{as}\quad N\to\infty\mbox{ .}
\]
\end{theorem}
\begin{proof}
In a complete $m$-ary tree of depth $N\geq 2$ the total number of nodes is $|V|=1+m+m^{2}+\cdots +m^{N-1}=(m^{N}-1)/(m-1)$. Recall the average path length of $G=(E,V)$ at a depth $N\geq 2$ is given by $\bar{P}(G)=\frac{S_{N}}{{ |V| \choose 2}}$ and as ${ |V| \choose 2}=O(m^{2N})$ we deduce from (\ref{eq:1}) that
\begin{eqnarray*}
\bar{P}(G) &amp; = &amp; \frac{m^{2N}}{{|V| \choose 2}}\left( \frac{N}{(m-1)^{2}}-\frac{m+1}{(m-1)^{3}}\right) + o(1)\\
           &amp; = &amp; \frac{2m^{2N}}{(m^{N}-1)^{2}-(m^{N}-1)(m-1)}\left( N-\frac{m+1}{m-1}\right) +o(1)\\
           &amp; \sim &amp; 2\left( N-1-\frac{2}{m-1}\right) \mbox{ as $N\rightarrow\infty$,}
\end{eqnarray*}
hence $\bar{P}(G)\sim D-\frac{4}{m-1}$ as $N\to\infty$.
\end{proof}

Clearly, by substituting $m=2$ into Theorem \ref{T1}, we arrive at the
asymptotic estimate $\bar{P}(G)\sim D-4$, when $G=(E,V)$ is a complete
binary tree, as first conjectured by T. Lewis \cite[p.\ 83]{lew}.

\vskip .5in

\begin{thebibliography}{9}
\bibitem{can} E. R. Canfield, R. W. Robinson, and D. H. Rouvray, Determination of the
Wiener molecular branching index for the general tree, {\em J. Comput. Chem.}
{\bf 6} (1985), 598--609.

\bibitem{dob} A. D. Dobrynin, R. Entringer, and I. Gutman, Wiener index of trees:
theory and applications, {\em Acta Applicandae Mathematicae} {\bf 66} (2001), 211--249.

\bibitem{lew} T. G. Lewis, {\em Network Science}, Wiley, 2009.

\bibitem{liu} H. Liu and X. Pan, On the Wiener index of trees with fixed diameter, {\em MATCH Commun. Math. Comput. Chem.} {\bf 60} (2008), 85--94.
\end{thebibliography}



\bigskip
\hrule
\bigskip

\noindent 2010 {\it Mathematics Subject Classification}:
Primary 05C30; Secondary 05C38.

\noindent \emph{Keywords: }average path length,
Wiener index, $m$-ary tree.

\bigskip
\hrule
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\vspace*{+.1in}
\noindent
Received January 23 2014;
revised version received May 7 2014.
Published in {\it Journal of Integer Sequences}, May 8 2014.

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\noindent
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