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.