  To each generic complex polynomial $p(z)$ there is associated a labeled
binary tree (here referred to as a "lemniscate tree") that encodes the
topological type of the graph of $|p(z)|$. The branching structure of the
lemniscate tree is determined by the configuration (i.e., arrangement in the
plane) of the singular components of those level sets $|p(z)|=t$ passing
through a critical point.
  In this paper, we address the question "How many branches appear in a typical
lemniscate tree?" We answer this question first for a lemniscate tree sampled
uniformly from the combinatorial class and second for the lemniscate tree
arising from a random polynomial generated by i.i.d. zeros. From a more general
perspective, these results take a first step toward a probabilistic treatment
(within a specialized setting) of Arnold's program of enumerating algebraic
Morse functions.
