  We consider in this paper the problem of discovering, via a traceroute
algorithm, the topology of a network, whose graph is spanned by an infinite
branching process. A subset of nodes is selected according to some criterion.
As a measure of efficiency of the algorithm, the Steiner distance of the
selected nodes, i.e. the size of the spanning sub-tree of these nodes, is
investigated. For the selection of nodes, two criteria are considered: A node
is randomly selected with a probability, which is either independent of the
depth of the node (uniform model) or else in the depth biased model, is
exponentially decaying with respect to its depth. The limiting behavior the
size of the discovered subtree is investigated for both models.
