  Tree-based protocols are ubiquitous in distributed systems. They are
flexible, they perform generally well, and, in static conditions, their
analysis is mostly simple. Under churn, however, node joins and failures can
have complex global effects on the tree overlays, making analysis surprisingly
subtle. To our knowledge, few prior analytic results for performance estimation
of tree based protocols under churn are currently known. We study a simple
Bellman-Ford-like protocol which performs network size estimation over a
tree-shaped overlay. A continuous time Markov model is constructed which allows
key protocol characteristics to be estimated, including the expected number of
nodes at a given (perceived) distance to the root and, for each such node, the
expected (perceived) size of the subnetwork rooted at that node. We validate
the model by simulation, using a range of network sizes, node degrees, and
churn-to-protocol rates, with convincing results.
