  Supertree methods are tree reconstruction techniques that combine several
smaller gene trees (possibly on different sets of species) to build a larger
species tree. The question of interest is whether the reconstructed supertree
converges to the true species tree as the number of gene trees increases (that
is, the consistency of supertree methods). In this paper, we are particularly
interested in the convergence rate of the maximum likelihood supertree.
Previous studies on the maximum likelihood supertree approach often formulate
the question of interest as a discrete problem and focus on reconstructing the
correct topology of the species tree. Aiming to reconstruct both the topology
and the branch lengths of the species tree, we propose an analytic approach for
analyzing the convergence of the maximum likelihood supertree method.
Specifically, we consider each tree as one point of a metric space and prove
that the distance between the maximum likelihood supertree and the species tree
converges to zero at a polynomial rate under some mild conditions. We further
verify these conditions for the popular exponential error model of gene trees.
