The average diameter of general tree structures

In this paper, we study the static behavior of distributed memory architecture with general tree structures. After defining and discussing the notion of average diameter, we first show that the average time it takes to send messages between any two arbitrary processors in a binary tree structure with n processors is θ(log n), through combinatorial analysis. We will also show that we can extend this result to general tree structures with n nodes, with a reasonable assumption, i.e., when m = o(n), where m is the maximum number of children any node could have. We believe that the results presented in this paper have captured some important inter-processor data transmission behavior of computers with distributed memory architectures, particularly those with (binary) tree structured inter-processor networks. As an application of combinatorial and asymptotic analysis, this paper solves yet another average path length problem, thus is also theoretically interesting. Finally, some of the techniques we have developed in this paper might also be useful in the study of similar problems.