Consider a binary tree topology for a packet-switching network. The root node connects to two other nodes. All intermediate nodes connect to one node in the direction toward the root, and two in the direction away from the root. At the bottom are nodes with just one link back toward the root. If there are 2N - 1 nodes, derive an expression for the mean number of hops per packet for large N, assuming that trips between all node pairs are equally likely. Hint: You will find the following equalities useful:

a

∞

i=1 Xi =

X 1 - X

; a

∞

i=1 iXi =

X

(1 - X)2