[Paper Review] Geometric comparison of phylogenetic trees with different leaf sets

Gillian Grindstaff, Megan Owen|arXiv (Cornell University)|Jul 11, 2018

Genomics and Phylogenetic Studies19 references3 citations

TL;DR

This paper introduces a geometric framework for comparing phylogenetic trees with different leaf sets by extending subtree topologies into a common higher-dimensional tree space (BHV space), enabling the computation of compatible supertrees and measuring compatibility through relaxed extension regions. The key contribution is a parameterized relaxation of extension spaces that ensures non-empty intersections and provides a quantitative measure of compatibility via the $ p_{\alpha} $ parameter.

ABSTRACT

The metric space of phylogenetic trees defined by Billera, Holmes, and Vogtmann, which we refer to as BHV space, provides a natural geometric setting for describing collections of trees on the same set of taxa. However, it is sometimes necessary to analyze collections of trees on non-identical taxa sets (i.e., with different numbers of leaves), and in this context it is not evident how to apply BHV space. Davidson et al. recently approached this problem by describing a combinatorial algorithm extending tree topologies to regions in higher dimensional tree spaces, so that one can quickly compute which topologies contain a given tree as partial data. In this paper, we refine and adapt their algorithm to work for metric trees to give a full characterization of the subspace of extensions of a subtree. We describe how to apply our algorithm to define and search a space of possible supertrees and, for a collection of tree fragments with different leaf sets, to measure their compatibility.

Motivation & Objective

• To address the challenge of comparing phylogenetic trees with non-identical taxa sets, which cannot be directly analyzed using standard BHV tree space methods.
• To develop a geometric method for extending subtrees into a common tree space to enable metric comparison and supertree construction.
• To characterize the subspace of possible extensions of a subtree in BHV space, particularly under relaxed compatibility conditions.
• To define and compute a compatibility measure between tree fragments with different leaf sets using the $ p_{\alpha} $ parameter.
• To enable the analysis of incomplete or fragmented phylogenetic data—such as gene trees with missing taxa—within a unified geometric framework.

Proposed method

• The method uses the tree dimensionality reduction map $ \Psi $ from $ \mathcal{T}^N $ to $ \mathcal{T}^\mathcal{L} $, and constructs its preimage $ \Psi^{-1} $ to lift subtrees into a common space.
• It defines extension regions $ E_{\mathbf{T}}^N(p_\alpha)_p $ as the set of trees in $ \mathcal{T}^N $ that extend given subtrees under a relaxed compatibility condition parameterized by $ p_\alpha \in [0,1] $.
• The relaxation allows for edge length perturbations bounded by $ p_\alpha \cdot w_e $, enabling non-empty intersection of extension regions even when strict compatibility fails.
• A linear program is formulated to compute the minimal $ p $ value for which the intersection of extension regions remains non-empty, providing the $ p_\alpha $ compatibility parameter.
• The method leverages orthant decomposition in BHV space to handle piecewise-linear structure and ensures that non-emptiness of extension regions is determined by orthant-wise feasibility.
• The approach generalizes prior combinatorial algorithms by adapting them to metric trees, allowing for continuous, geometric compatibility analysis.

Experimental results

Research questions

• RQ1 How can phylogenetic trees with different leaf sets be meaningfully compared in a geometric framework when standard BHV space methods require identical taxa sets?
• RQ2 What is the geometric characterization of the set of all possible extensions of a subtree into a larger tree space with a common set of taxa?
• RQ3 Can a relaxed compatibility condition be defined that ensures non-empty intersection of extension regions even when trees are not strictly compatible?
• RQ4 How can the $ p_\alpha $ parameter be computed efficiently, and what does it represent in terms of metric deviation from strict compatibility?
• RQ5 What is the relationship between the $ \alpha $-relaxation and $ p_\alpha $-relaxation in terms of extension region non-emptiness and geometric stability?

Key findings

• The $ p_\alpha $ parameter is defined as the minimal relaxation level for which the intersection of extension regions remains non-empty, and it is computable via a linear program.
• For any tree set $ \mathbf{T} $, the extension region $ E_{\mathbf{T}}^N(p_\alpha)_p $ is non-empty if and only if at least one orthant contains a feasible solution, and $ p_\alpha $ equals the minimal such $ p $ across orthants.
• The cone point in BHV space is included in the relaxed extension region when $ p = 1 $, guaranteeing non-emptiness for $ p_{\alpha} \leq 1 $, which ensures the method is always applicable.
• When $ p_{\alpha} < 1 $, the cone point is excluded from the extension region, preserving geometric fidelity to the original tree structure.
• The $ p_\alpha $ parameter is normalized to $[0,1]$, with $ p_{\alpha} = 0 $ indicating strict compatibility and $ p_{\alpha} = 1 $ indicating maximal relaxation.
• The method ensures that $ N_\alpha \subset E_{\mathbf{T}}^N(\alpha) \subset E_{\mathbf{T}}^N(p)_p $ for $ \alpha < \frac{p}{\log_2 N} \cdot \min_{e \in \mathbf{T}} w_e $, linking the two relaxation schemes.