Figure 1.

Gene tree landscape. a) Each point within the landscape corresponds to a gene tree T_x, whose optimality can be measured through its likelihood L_x (height) and its reconciliation cost c_x (color). The ML tree T_ML is located at the peak of this landscape but may have a high cost. Rearranging T_ML to a nearby tree T_x can result in a negligible decrease in likelihood (δ_x = L_ML − L_x < δ_thr) while simultaneously reducing the tree cost (c_x < c_ML), thus producing a more congruent gene tree that is statistically equivalent to the ML tree. TreeFix utilizes this basic idea by balancing the 2 optimality criteria to return an optimal tree T^* for which δ^* is negligible and c^* is minimal. b) The landscape for a simulated gene family shows a wide range of likelihood and cost values. In this instance, TreeFix searched over 3560 gene trees of the 8.2 × 10²¹ possible unrooted topologies (number of genes = 21). Although most trees have statistically worse likelihoods compared with the ML tree (×), a subset of high likelihood trees are statistically equivalent (circles). As the search progresses, the search space generally moves toward the top-left, corresponding to topologies with high likelihood and low duplication–loss cost (enlarged at right, accepted trees per iteration shown as squares). In this case, TreeFix has rearranged T_ML (beige triangle) to produce a new optimal tree T^* (purple triangle) with equivalent likelihood and lower cost. Note that T^* is incorrect because the true tree T_true (black triangle) has a slightly higher duplication–loss cost. (Likelihoods were computed with ϵ = 2.)