Summary: 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 $D - 4/(m - 1)$, 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.

average path length, Wiener index, m-ary tree