A Vectorial Tree Distance Measure

A vectorial distance measure for trees is presented. Given two trees, we align the trees from their centers outwards, starting from the root-branches, to make the next level as similar as possible. The algorithm is recursive; condition on the alignment of the root-branches we align the sub-branches, thereafter each alignment is conditioned on the previous one. We define a minimal alignment under a lexicographic order which follows the intuition that the differences between the two trees closer to their cores dominate their differences at a higher level. Given such a minimal alignment, the difference in the number of branches calculated at any level defines the entry of the distance vector at that level. We compare our algorithm to other well-known tree distance measures in the task of clustering sets of phylogenetic trees. We use the TreeSimGM simulator for generating stochastic phylogenetic trees. The vectorial tree distance can successfully separate symmetric from asymmetric trees, and hierarchical from non-hierarchical trees.

Probabilistic Species Tree Distances: Implementing the Multispecies Coalescent to Compare Species Trees Within the Same Model-Based Framework Used to Estimate Them

Despite the ubiquitous use of statistical models for phylogenomic and population genomic inferences, this model-based rigor is rarely applied to post hoc comparison of trees. In a recent study, Garba et al. derived new methods for measuring the distance between two gene trees computed as the difference in their site pattern probability distributions. Unlike traditional metrics that compare trees solely in terms of geometry, these measures consider gene trees and associated parameters as probabilistic models that can be compared using standard information theoretic approaches. Consequently, probabilistic measures of phylogenetic tree distance can be far more informative than simply comparisons of topology and/or branch lengths alone. However, in their current form, these distance measures are not suitable for the comparison of species tree models in the presence of gene tree heterogeneity. Here, we demonstrate an approach for how the theory of Garba et al. (2018), which is based on gene tree distances, can be extended naturally to the comparison of species tree models. Multispecies coalescent (MSC) models parameterize the discrete probability distribution of gene trees conditioned upon a species tree with a particular topology and set of divergence times (in coalescent units), and thus provide a framework for measuring distances between species tree models in terms of their corresponding gene tree topology probabilities. We describe the computation of probabilistic species tree distances in the context of standard MSC models, which assume complete genetic isolation postspeciation, as well as recent theoretical extensions to the MSC in the form of network-based MSC models that relax this assumption and permit hybridization among taxa. We demonstrate these metrics using simulations and empirical species tree estimates and discuss both the benefits and limitations of these approaches. We make our species tree distance approach available as an R package called pSTDistanceR, for open use by the community.

Unexplained Relationships of Height-Diameter of Three Tree Species in a Tropical Forest.

Associations between bivariate variables relative to the unexplained relationships of height-Dbh (diameter at breast height) models were investigated. Seven permanent sample plots measuring 40m by 250m at Omo Forest Reserve were used to assess the relationships between height and diameter at breast height of three tree species as affected by the variables of neighbouring trees. The result showed differences in the coefficient of determination of the bivariate models and multivariate models. The models arrived at for each of the species and for the bivariate models are: Scottelia coriaceae: Ht = 2.59 + 0.473D+ 0.0012D2 for 5cm ≤ D ≤100cm (R2 = 0.70) Sterculia rhinopetala: Ht = 5.96 + 0.467D+ 0.00296D2 for 5cm ≤ D ≤100cm (R2 = 0.77) Strombosia pustulata: Ht = 2.02 + 0.722D+ 0.00581D2 for 5cm ≤ D ≤ 60cm (R2 = 0.81) Where both Ht and D are height and Diameter at breast height. While on the other hand the multivariate models that considered the effect of neighbouring trees are: Scottelia coriaceae: 2 1 2 3 4 Ht = 3.74 + 0.41x −1.14x + 0.205x +1.278x (R = 0.723) Sterculia rhnopetala: 2 1 2 3 4 Ht = 6.18 + 0.2601x +1.163x + 0.438x − 0.442x (R = 0.608) Strombosia pustalata: 2 1 2 3 4 Ht = 6.84 + 0.399x − 0.318x − 0.138x − 0.838x (R = 0.650) x1 = diameter at breast height, x2 = Mean neighbouring tree distance, x3 = Frequency of the neighbouring tree and x4 = Position of the crown.