# Phylogenetic Vignettes: Tree Rearrangements

As I mentioned in the previous post, I am taking some time off to figure out which direction to pursue next in the blog post series. However, that does not mean that the whole series itself is going into hiatus. In the meantime, I am going to cover a few other topics in phylogenetics. Additionally, I will cross post some of the one-off posts on Dr. Treangen’s lab website. Now, with logistics out of the way let’s get into tree rearrangements.

## Introduction

Whenever we are faced with a problem in combinatorial optimization it is often convenient (or even necessary as the majority of interesting problems turn out to be NP-hard) to come up with some heuristic methods. In particular, we often want to explore a “neighborhood” of an object and locate optimal candidates within this neighborhood after which the search can continue from the new local optima. In order to talk about neighborhoods we need to define a metric on the space of objects we are working with (we talked about metric spaces on phylogenetic trees in several previous posts, but in this case we are focusing solely on the tree topology and associated finite metric spaces). In the context of a search problem it is useful to think about the neighborhood of an object as a set of objects that can be obtained from the initial object via some fixed operation. In phylogenetics such operations on trees are typically called tree rearrangements (Felsenstein, 2004). In this post we will take a look at three classical tree rearrangement operations, but keep in mind that there are other ways to mutate trees which lead to different geometries on the tree space.

In the next three sections we will introduce in detail each of the rearrangement operations in the order of increasing neighborhood sizes considered and then briefly discuss the relationships between these three classes, as well as some implications for computational complexity of related problems.

## Nearest neighbor interchange

Nearest neighbor interchange (NNI) operation on a tree consists of picking an internal edge (i.e. an edge that is not incident to a leaf) deleting it along with the four edges incident to the vertices of the original edge, after obtaining four components as the result of the deletion there are three distinct ways of reconnecting them back out of which one yields the original tree. It is easy to see that for an unrooted bifurcating tree on n taxa (i.e. a tree with n leaves) there is a total of n-3 internal edges, and hence a total of 2(n-3) NNI-neighbors. We also note that the operation in this classical sense is defined for bifurcating trees (hence the guarantee that the NNI yields for connected components after the edge removal step). The figure below provides an example of NNI and the resulting tree neighbors.

For a small value of n we can reasonably visualize the adjacency structure on the tree space induced by the NNI operation. Namely we can identify non-isomorphic (read: distinct from the point of view of the properties we care about) labeled tree topologies as vertices and let two vertices be connected by an edge iff we can obtain one topology from another via a single NNI.

In general a phylogenetic tree encountered (or I guess inferred) in the wild might not be bifurcating, so it is quite natural to ask how does one generalize the notion of NNI to an arbitrary labeled unrooted tree. At the core of this operation is the process of picking a distinguished edge and then swapping components resulting from an edge deletion operation. Thus, in the case of a multifurcating tree we simply can do the same operation but with a larger set of neighbors being generated. An example is provided in the figure below.

There is some amount of work on the NNI operations and NNI induced metric on phylogenetic trees (Hon and Lam, 1999), although the area tends to feel relatively sparse in terms of the research done.

## Subtree prune and regraft

Subtree prune and regraft (SPR) operation on a tree consists of snipping off a branch (which can be an internal or external edge) and then inserting the snipped off subtree onto one of the edges of the remaining tree. This operation is quite well illustrated by its name, especially if you ever had to deal with grafts on actual trees (my grandfather loved to experiment with fruit tree grafting, which to his credit managed to give us a wonderful apricot half-tree on a wild apricot that used to grow in the garden). The figure below is an example of a SPR operation performed on a tree.

It is easy to see that SPR operation results in a larger neighborhood set than the NNI operation. It also useful to note that any tree T’ that can be obtained from a tree T via a single NNI can also be obtained via a single SPR. In other words, if we define the set NNI(T) of trees that can be obtained from T via a single NNI, and the set SPR(T) of trees that can be obtained from T via a single SPR, then the following inclusion holds: NNI(T)SPR(T) (Maddison, 1991). In the spirit of ancient geometers instead of a formalized proof, we will provide the following figure and leave it up to the reader to convince themselves that the inclusion above is indeed true.

Similarly to the NNI case we can compute the exact number of neighbors under the SPR operation for an unrooted bifurcating binary tree on n taxa. The size of this neighborhood similarly to the NNI case is independent of the topology of the tree T and is equal to 2(n-3)(2n-7) with a detailed counting argument provided by Allen and Steel, 2001 on p. 4.

We also note that the way in which we graft a subtree back in will always create an internal node of degree 3, so without a modification to this operation the question of generalizing it to multifurcating trees can be ill-posed.

## Tree bisection and reconnection

Tree bisection and reconnection (TBR) operation is the most expansive operation among the three we are considering in this post, in the sense that it yields the largest neighborhoods. TBR consists of splitting the original tree T into two subtrees T’ and T” along a branch and then reconnecting them by joining any two branches of T’ and T” via an edge (in case if one of the subtrees is just a leaf the leaf is simply joined back to the tree via a new branch; in all cases vertices might have to be added to maintain a proper bifurcating tree). The figure below illustrates a TBR operation being performed.

The inclusion we observed for the NNI and SPR neighborhoods extends further and the following holds: NNI(T)SPR(T)TBR(T) (Maddison, 1991). Furthermore, by noting that any TBR can be thought of as a two step process where we first re-attach T” onto the specified edge of T’ and then reposition the attached T’ in order to get correct edge match, we can show that any TBR operation can be replicated by at most two successive SPR operations.

Finally, we note that unlike NNI and SPR, the TBR neighborhood size does depend on the topology of the tree T. However, and upper bound on the size of the neighborhood can be computed and is given by (2n-3)(n-3)2 (Allen and Steel, 2001, p.5).

## Induced metrics and computational (in)tractability

We already noted in the introduction that the tree rearrangement operations induce corresponding metrics on the space of phylogenetic trees (more precisely in our current case: the space of unrooted bifurcating tree topologies). We will denote these metrics dNNI(T, T’), dSPR(T, T’), and dTBR(T, T’), respectively. From the inclusion relation on the neighborhoods it immediately follows that for any two unrooted bifurcating trees we have the following inequality: dTBR(T, T’)dSPR(T, T’)dNNI(T, T’). Furthermore, from our observation in the previous section we also can conclude that dSPR(T, T’)dTBR(T, T’).

Although we have provided an explicit example of the adjacency graph under NNI metric for the space of unrooted bifurcating trees on 5 taxa, it is not immediately obvious that in the general case the NNI adjacency graph will be connected (i.e. we are not a priori guaranteed to have a connected metric space). However, it turns out that indeed the adjacency graph under the NNI metric is indeed connected, and hence the metric space induced by dNNI(T, T’) is connected. It is easy to see that as a corollary we also get connectedness for the SPR and TBR spaces.

Since the metric spaces are connected and finite we can ask what are their respective diameters (i.e. the smallest distance between two points furthest apart in the space; this is the same definition as the graph diameter). An interesting non-trivial bound on the diameter of the NNI space was derived by Li et al., 1996: authors show that the diameter of the NNI adjacency graph (which we will denote as ΔNNI) is bounded below and above by nlogn terms. More precisely, the following inequality holds: $\frac{n}{4}\log_2 n - o(n\log n)\leq \Delta_{NNI}\leq n\log_2 n + O(n)$. Allen and Steel, 2001 extend this result to the SPR and TBR cases giving the following inequalities: (a) n/2-o(n)ΔSPRn-3; (b) n/4-o(n)ΔTBRn-3.

However, knowing the diameter of a space does not imply that we can easily compute the distances within it (i.e. shortest paths via NNI, SPR or TBR operations from one tree to another). In fact, the problems of computing NNI, SPR or TBR distance between two arbitrary unrooted bifurcating trees are all NP-hard (NNI: DasGupta et al., 2000; SPR: Bordewich and Semple, 2005, Hickey et al., 2008; TBR: Hein et al., 1996). Allen and Steel, 2001 show that the SPR distance is fixed-parameter tractable, but in general the fixed-parameter tractability results in algorithms that can work efficiently only on small trees or pairs of trees which are relatively close to each other (for a more principled discussion see Whidden and Matsen, 2018). In general, it appears to be a rare case for a metric that is biologically interpretable to be efficiently computable, and vice-a-versa (take for example the Robinson-Foulds distance which is efficiently computable, but not biologically well grounded). There are some recent attempts at developing variants of tree rearrangement operations that have both biological interpretability and are computationally tractable (Collienne and Gavryushkin, 2021).

While we could possibly discuss other tree rearrangement operations or explore the discrete geometries arising from the ones we mentioned in more detail, such work would likely require multiple posts. Thus, we will wrap up here and invite the reader to continue the exploration via following the links provided in the references section.

## Data and code availability

All illustrations with the exception of the one from Felsenstein, 2004 were made with draw.io. No additional materials or code were used in this post.

## What’s next

I am still deliberating on the exact direction to take for the next stretch of posts, so in the meantime I am intending to continue the one-off posts where I pick a random, but interesting to me, topic in mathematical phylogeny and try to write a coherent text about it. As per usual, if you are interested in me writing about a specific topic do not hesitate to reach out.

## References

Allen, Benjamin L., and Mike Steel. “Subtree transfer operations and their induced metrics on evolutionary trees.” Annals of combinatorics 5, no. 1 (2001): 1-15. https://doi.org/10.1007/s00026-001-8006-8

Bordewich, Magnus, and Charles Semple. “On the computational complexity of the rooted subtree prune and regraft distance.” Annals of combinatorics 8, no. 4 (2005): 409-423. https://doi.org/10.1007/s00026-004-0229-z

Collienne, Lena, and Alex Gavryushkin. “Computing nearest neighbour interchange distances between ranked phylogenetic trees.” Journal of Mathematical Biology 82, no. 1 (2021): 1-19. https://doi.org/10.1007/s00285-021-01567-5

Felsenstein, Joseph. Inferring phylogenies. Vol. 2. Sunderland, MA: Sinauer associates, 2004.

Gupta, Bhaskar Das, Xin He, Tao Jiang, Ming Li, and John Tromp. “On computing the nearest neighbor interchange distance.” In Discrete Mathematical Problems with Medical Applications: DIMACS Workshop Discrete Mathematical Problems with Medical Applications, December 8-10, 1999, DIMACS Center, vol. 55, p. 125. American Mathematical Soc., 2000. PDF on author’s page

Hein, Jotun, Tao Jiang, Lusheng Wang, and Kaizhong Zhang. “On the complexity of comparing evolutionary trees.” Discrete Applied Mathematics 71, no. 1-3 (1996): 153-169. https://doi.org/10.1016/S0166-218X(96)00062-5

Hickey, Glenn, Frank Dehne, Andrew Rau-Chaplin, and Christian Blouin. “SPR distance computation for unrooted trees.” Evolutionary Bioinformatics 4 (2008): EBO-S419. https://doi.org/10.4137/EBO.S419

Hon, Wing-Kai, and Tak-Wah Lam. “Approximating the nearest neighbor interchange distance for evolutionary trees with non-uniform degrees.” In International Computing and Combinatorics Conference, pp. 61-70. Springer, Berlin, Heidelberg, 1999. https://doi.org/10.1007/3-540-48686-0_6

Li, Ming, John Tromp, and Louxin Zhang. “On the nearest neighbour interchange distance between evolutionary trees.” Journal of Theoretical Biology 182, no. 4 (1996): 463-467. https://doi.org/10.1006/jtbi.1996.0188

Maddison, David R. “The discovery and importance of multiple islands of most-parsimonious trees.” Systematic Biology 40, no. 3 (1991): 315-328. https://doi.org/10.1093/sysbio/40.3.315

Whidden, Chris, and Frederick A. Matsen. “Calculating the unrooted subtree prune-and-regraft distance.” IEEE/ACM transactions on computational biology and bioinformatics 16, no. 3 (2018): 898-911. https://doi.org/10.1109/TCBB.2018.2802911

# Phylogenetic Vignettes: Owen-Provan algorithm

## Important announcement

I am currently deciding on the direction this post series will take after next week. Please check out the What’s next section and let me know if you have any preferences among described routes.

## Recap

In the previous post (following Gavryushkin and Drummond, 2016) we have defined a geometry on the space of phylogenetic ultrametric trees and called the resulting metric space 𝛕-space. We also briefly mentioned the geometry on the space of trees defined by Billera, Holmes and Vogtmann, 2001, and referred to the resulting construction as BHV space.

In this post we will provide a brief overview of the algorithm presented in Owen and Provan, 2010, which allows us to compute the geodesics in 𝛕-space and BHV space in polynomial time (in the number of the tree leaves/taxa). For the ease of exposition we will delegate detailed overview of the BHV space to a later post. Hence, in what follows we will assume without the proof that BHV space is a CAT(0) complex and trees in BHV space are parametrized by the internal branch lengths with different orthants corresponding to different tree topologies.

## Goal

As we mentioned before one of the key properties we want from a metric space structure on the tree space is the efficient computation of geodesics. Clearly given any finite metric space (complexes constructed in BHV or 𝛕-spaces are not finite, but the topologies are and the problem of distance within a fixed orthant determined by topology is easy to solve) we can simply consider all possible paths and then pick the one minimizing the length function. However, such approach yields an algorithm that will fail to finish computations even for modestly sized spaces.

Thus, our goal is to have an efficient algorithm that can find shortest path in polynomial time with respect to the number of leaves our trees can have.

## CAT(0) properties

Since both BHV and 𝛕-spaces are CAT(0) we can leverage nice properties of the non-positively curved geometries to get an idea for an efficient algorithm. The key idea behind the algorithm comes from the following lemma (Lemma 2.1 in Owen and Provan, 2010, and Proposition 1.4, p. 160 in ).

Lemma 1. In CAT(0) space, every local geodesic is a geodesic.

Proof. Consider a local geodesic $\Gamma_\varepsilon$ between points $P$ and $Q$. Choose a set of points $P=P_0,P_1,...,P_k=Q$ such that $L(\Gamma_\varepsilon(P_i, P_{i+1}))\leq \varepsilon/2$. Note that it follows from the definition of a local geodesic that $\Gamma_\varepsilon(P_{i-1}, P_{i+1})$ is a geodesic. Now, assume by induction that $\Gamma_\varepsilon(P_0, P_j)$ is a geodesic for all $j\leq i$. Hence, we have $L(\Gamma_\varepsilon(P_0, P_{i-1}))=d(P_0, P_{i-1})$ by the induction hypothesis and $L(\Gamma_varepsilon(P_{i-1},P_{i+1}))=d(P_{i-1}, P_{i+1})$ as noted above. Consider a corresponding model triangle $p_0p_{i-1}p_{i+1}$ in Euclidean space (if you forgot what a model triangle is check: Bridson and Haefliger, 2013 p. 158 or previous post on CAT spaces). Let $p_i$ be a point on $p_{i-1}p_{i+1}$ such that $|p_{i-1}p_i|=d(P_{i-1}, P_i)$. Since we are working in a CAT(0) space it follows that $d(P_0, P_i)\leq |p_0p_i|$. On the other hand, by induction hypothesis we have $d(P_0, P_i) = d(P_0, P_{i-1}) + d(P_{i-1}, P_i) = |p0p_{i-1}|+|p_{i-1}p_i|$. Combining this result with triangle inequality gives us $|p0p_{i-1}|+|p_{i-1}p_i|=|p_0p_i|$, and hence $p_{i-1}\in p_0p_i$ and therefore $p_{i-1}\in p_0p_{i+1}$. From here we can conclude that $\Gamma_\varepsilon(P_0, P_{i+1}$ is a geodesic and hence $\Gamma_\varepsilon$ is a geodesic.

Thus, the main idea used in the paper is to start with any path between two trees (consider the path from a tree to the star tree and then back to target tree, this path called cone path is a common idea in the analyses of tree spaces; recall: Robinson-Foulds metric) and iteratively check whether it’s a local geodesic refining the path in cases when that’s not the case.

## Path space geodesics

Following Owen and Provan, 2010 we will assume that the trees $T, T^\prime$ between which we are computing geodesics have distinct edges. We will also call two sets of edges $A\subseteq E(T)$ and $B\subseteq E(T^\prime)$ compatible if all pairs of splits associated with these edges are compatible, or equivalently if the union $A\cup B$ defines a tree. In other words, compatible sets of edges produce “intermediate” trees between $T$ and $T^\prime$. Now, for a pair of partitions $\mathcal{A}=\{A_1,...,A_k\}$, $\mathcal{B}=\{B_1,...,B_k\}$ of $E(T)$ and $E(T^\prime)$, respectively, we can define an associated path space given that the following property holds: for every $i>j$, $A_i$ and $B_j$ are compatible. To construct the path space consider the collection of orthants $\mathcal{P}(T, T^\prime)=\cup_{i=1}^k O_i$ given by $O_i=O(B_1\cup ...\cup B_i\cup A_{i+1}\cup ...\cup A_k)$, where $O(T)$ is the mapping of a tree to the corresponding orthant. We will call the corresponding partition pair $(\mathcal{A}, \mathcal{B})$ the support, and the shortest path between $T$ and $T^\prime$ in $\mathcal{P}$ the path space geodesic. These definitions make the geodesic in the tree space a more concrete and constructable object, which of course is important for any algorithmic approach. Furthermore, the following theorem from Billera, Holmes and Vogtmann, 2001 (see Proposition 4.1) guarantees that the path space geodesics are recovering the true geodesics.

Theorem 1. For disjoint trees $T, T^\prime$ the geodesic is a path space geodesic for some choice of $\mathcal{P}(T, T^\prime)$.

We will also need the following two results, the first of which is proven in Owen, 2011 and the second in Owen and Provan, 2010 (Theorem 2.5).

Theorem 2. Let $\Gamma$ be a geodesic between $T$ and $T^\prime$. Then there exist partitions $\mathcal{A}=\{A_1,...,A_k\} and$latex \mathcal{B}=\{B_1,…,B_k\} of $E(T)$ and $E(T^\prime)$, respectively, such that $(\mathcal{A}, \mathcal{B})$ (a) form a support of path space in which $\Gamma$ is a path space geodesic, and (b) the following condition is satisfied:

$\frac{\parallel A_1\parallel}{\parallel B_1\parallel}\leq \frac{\parallel A_2\parallel}{\parallel B_2\parallel}\leq\cdots\frac{\parallel A_k\parallel}{\parallel B_k\parallel}$.

A path space satisfying the above condition for its support is referred to as a proper path space, and the corresponding path space geodesic is called a proper path.

Theorem 3. A proper $(T, T^\prime)$-path $\Gamma$ with support $(\mathcal{A}, \mathcal{B})$ is a geodesic iff for each pair of edge sets $(A_i, B_i)$ in the support there is no nontrivial partition $C_1\cup C_2=A_i$, $D_1\cup D_2=B_i$ such that $C_2$ is compatible with $D_1$ and $\frac{\parallel C_1\parallel}{\parallel D_1\parallel}\leq \frac{\parallel C_2\parallel}{\parallel D_2\parallel}$.

Theorem 3 gives us the final ingredient needed to build out an algorithm for finding BHV space geodesics. Namely, partitions of edge sets give us a way to construct proper paths and the condition on local improvement of such proper paths guarantees that we achieve a geodesic.

## Algorithm overview

As we mentioned earlier the natural choice for the starting path is the cone path between the two trees $T$ and $T^\prime$, more formally it’s a proper path space defined by the support $(\mathcal{A}=\{E(T)\}, \mathcal{B}=\{E(T^\prime)\})$ and the corresponding path is given by uniform contraction of all edges which results in star tree and then a follow up uniform decontraction. Now, the algorithm will proceed iteratively at each step checking whether the condition in Theorem 3 is satisfied and if not proposing a new support and a new proper path.

In order to facilitate an efficient solution for the iterative step, the problem of checking the condition of Theorem 3 and finding a new path can be recasted as an independent set problem over a bipartite graph. Namely, we will construct the incompatibility graph $G(A, B) between two edge sets$latex A\subseteq E(T), B\subseteq E(T^\prime)$as a bipartite graph with vertices given by $A\cup B$ and edges given by the pairs $e\in A, f\in B$ where the splits associated with $e$ and $f$ are incompatible. It is easy to see that for any $A\subseteq E(T), B\subseteq E(T^\prime)$ compatibility of $A$ and $B$ is equivalent to $A\cup B$ forming an independent set in$G(E(T), E(T^\prime))$. Thus, the condition from Theorem 3, can now be recasted as follows: Given two edge sets$A\subseteq E(T), B\subseteq E(T^\prime)$does there exist a pair of partitions $A=C_1\cup C_2$, $B=D_1\cup D_2$ such that the vertex set $C_2\cup D_1$ is independent in $G(A, B)$ and $\frac{\parallel C_1\parallel}{\parallel D_1\parallel}\leq \frac{\parallel C_2\parallel}{\parallel D_2\parallel}$. In particular, a proper path with support $(\mathcal{A}, \mathcal{B})$ is geodesic iff no edge set pair $(A_i\in\mathcal{A},B_i\in\mathcal{B})$ satisfies the above condition. Now, since the complement of an independent set is a vertex cover we can reformulate the above problem as a minimum weight (by transforming the inequality condition) vertex cover problem, which can be solved on bipartite graphs in $O(n^3)$. Further analysis of the algorithm steps yields that the total runtime to find the geodesic is $O(n^4)$. For the exact details of the proof of correctness of the algorithm and the runtime I suggest reading the original manuscript by Owen and Provan, 2010. ## Data/code availability This post has no associated custom code or data. Those looking for an implementation of the algorithm described should check out Megan Owen’s webpage or GitHub repo. ## What’s next This post concludes the beginning of our exploration of the geometry of the space of phylogenetic trees. There are several ways we can proceed from here. First, we can finally start tackling the foundational paper of Billera, Holmes and Vogtmann, 2001. This path will likely require us to spend a few posts dissecting original BHV space and some associated results from metric geometry and CAT(0) spaces. We then can return to the paper of Gavryushkin and Drummond, 2016 and explore the t-space construction in a way similar to our exploration of the 𝛕-space. Finally, we can then continue our journey by catching up to more recent work by Megan Owen et al. on defining and computing statistics in BHV space (Brown and Owen, 2020), comparing phylogenetic trees with different leaf sets (Grindstaff and Owen, 2018), and extending the framework to phylogenetic cactuses (a type of phylogenetic network, Huber et al., 2021). Second, we can instead pivot completely and ask a general question of what other geometries one can consider on the space of phylogenetic trees. In particular, we can take a look at algebraic geometry inspired view of the phylogenetic tree space as a tropical geometry (Monod et al., 2018), and an information geometry angle which gives rise to wald space (pun credit to the authors) of the phylogenetic trees (Garba et al., 2021). I will most likely skip the category theoretic angle for the topic (Baez and Otter, 2015), since frankly speaking I have no desire to introduce the notion of an operad. Third, we can temporarily halt the process of investigating deeper into the various geometries on the tree space and ask more broadly what are geometries of interest in the case of phylogenetic networks. This route will require us to go back to some definitions of the phylogenetic network and related concepts, and will likely demand at least some amount of rehashing of the biological context. Since, I am still debating which route is of the most interest to me at the moment, next week’s post will likely be sparser than usual, as I am figuring out which rabbit hole to pick as the next destination. However, if you actually would prefer me to pick one of these three directions do not hesitate to comment or reach out to me via other means of contact. ## References Baez, John C., and Nina Otter. “Operads and phylogenetic trees.” arXiv preprint arXiv:1512.03337 (2015). arXiv:1512.03337 Billera, Louis J., Susan P. Holmes, and Karen Vogtmann. “Geometry of the space of phylogenetic trees.” Advances in Applied Mathematics 27, no. 4 (2001): 733-767. https://doi.org/10.1006/aama.2001.0759 Bridson, Martin R., and André Haefliger. Metric spaces of non-positive curvature. Vol. 319. Springer Science & Business Media, 2013. https://doi.org/10.1007/978-3-662-12494-9 Brown, Daniel G., and Megan Owen. “Mean and variance of phylogenetic trees.” Systematic biology 69, no. 1 (2020): 139-154. https://doi.org/10.1093/sysbio/syz041, arXiv:1708.00294 Garba, Maryam Kashia, Tom MW Nye, Jonas Lueg, and Stephan F. Huckemann. “Information geometry for phylogenetic trees.” Journal of Mathematical Biology 82, no. 3 (2021): 1-39. https://doi.org/10.1007/s00285-021-01553-x, arXiv:2003.13004 Gavryushkin, Alex, and Alexei J. Drummond. “The space of ultrametric phylogenetic trees.” Journal of theoretical biology 403 (2016): 197-208. arXiv:1410.3544 Grindstaff, Gillian, and Megan Owen. “Geometric comparison of phylogenetic trees with different leaf sets.” arXiv preprint arXiv:1807.04235 (2018). arXiv:1807.04235 Huber, Katharina T., Vincent Moulton, Megan Owen, Andreas Spillner, and Katherine St John. “The space of equidistant phylogenetic cactuses.” arXiv preprint arXiv:2111.06115 (2021). arXiv:2111.06115 Monod, Anthea, Bo Lin, Ruriko Yoshida, and Qiwen Kang. “Tropical geometry of phylogenetic tree space: a statistical perspective.” arXiv preprint arXiv:1805.12400 (2018). arXiv:1805.12400 Owen, Megan. “Computing geodesic distances in tree space.” SIAM Journal on Discrete Mathematics 25, no. 4 (2011): 1506-1529. arXiv:0903.0696 Owen, Megan, and J. Scott Provan. “A fast algorithm for computing geodesic distances in tree space.” IEEE/ACM Transactions on Computational Biology and Bioinformatics 8, no. 1 (2010): 2-13. https://doi.org/10.1109/TCBB.2010.3 # Phylogenetic Vignettes: 𝛕-space In the previous post we have introduced the notion of a CAT(0) space, and briefly talked about some of the properties of CAT(0) spaces, and in particular cubical complexes. In this post, we will take this knowledge a step further into the applied direction of phylogeny. Following Gavryushkin and Drummond, 2016 we will define a metric on the space of ultrametric phylogenetic trees, show that the resulting space is CAT(0), and then discuss some of the implications of these results. Similarly to the previous post, we will have quite a bit of mathematical notation to deal with, so I will attempt to use visual aids whenever possible to improve exposition. ## Motivation One way of defining a metric on a space of phylogenetic trees is to embed the tree space $\mathcal{T}$ into some metric space $\mathcal{M}$. Thus, by associating every tree in the tree space with some point in the chosen metric space, we induce a pseudometric on tree space. However, such embeddings/parameterizations are not guaranteed to be “nice” with respect to the questions we aim to answer. Namely, not all embeddings are injective (multiple distinct trees can end up being mapped to the same point in $M$, hence in general we induce a pseudometric rather than a metric via embedding) although in the case of Gavryushkin and Drummond, 2016 the embedding has to be injective by definition. Additionally, not all embeddings are surjective (meaning that a path in $M$ might contain non-tree associated points) which creates a plethora of problems ranging from potential multiple distance minimizing solutions to the lack of tree space midpoints. Thus, it is natural to formulate a list of desiderata for the embedding $p:\mathcal{T}\to\mathcal{M}$ that would enable fruitful analyses. From now onwards (unless otherwise specified) we will assume that the embedding $p$ is injective. In order to make continuous distributions behave correctly under the pullback into the tree space, we have to require the image of the embedding to be path-connected. Note, that having a path connected image is also intuitively “nice” as the continuous tree transformations would result in a path being traced out in the model metric space. Hence, we have: (D1) The set $\mathrm{Image}(p)$ is path-connected in $\mathcal{M}$. However, being path-connected does not guarantee that the shortest path between the two points will lie within the image. Hence, we also want: (D2) The set $\mathrm{Image}(p)$ is convex in $\mathcal{M}$. We note that the injectivity criterion forces a lower bound on the dimension of $\mathcal{M}$. However, if we want to define probability measure on $\mathcal{M}$ that can be meaningfully pulled back onto $\mathcal{T}$ we need to ensure that: (D3) $\mathrm{Image}(p)$ has the same dimension as $\mathcal{M}$. Next, for statistical analyses to stay sound we need to ensure uniqueness of the shortest paths. This is the point in our desiderata list where you might have a brief flashback to the CAT(0) space discussion from previous week. (D4) The space $\mathcal{M}$ is uniquely geodesic. Finally, since we are ultimately tackling these questions from the computer science viewpoint, we have the last desiderata that ties this into the computational realm. Namely, we want the following: (D5) Geodesics in $\mathcal{M}$ are computable. In reality we probably want an even stronger version of (D5), which is: (D5′) Geodesics in $\mathcal{M}$ are efficiently (i.e. polynomial time) computable. It’s worth remarking that in a general metric space we do not necessarily have (D5) with a simple example being the halting problem reduction to shortest paths in an infinite graph. It is also important to realize that these desiderata do not guarantee existence of such embeddings, and are rather criteria for discerning more or less useful parameterizations of the tree space. ## A quick aside In general, we can attempt to parametrize the whole tree space (that’s precisely what Billera, Holmes and Vogtmann, 2001 do in their paper, which lies at the foundation of the connection between the geometry of non-positive curvature spaces and phylogenetic trees; we will refer to this space as the BHV space going forward) or we might focus on a specific subclass of trees such as ultrametric phylogenetic trees (recall that a tree is ultrametric is the distance from root to every leaf is the same). The caveat is that a parametrization that enjoys our desiderata (D1)-(D5′) for the whole tree space can fail to do so when restricted to the space of ultrametric trees. This is precisely the case for the BHV space parametrization. Hence, it can be of interest (motivated by the evolutionary models context) to explore parametrizations designed specifically for the space of the ultrametric trees. This is precisely what is done in Gavryushkin and Drummond, 2016, and what we are aiming to do in this post. ## 𝛕-space We now will construct one version of a space of ultrametric trees called the 𝛕-space. To proceed consider an ultrametric tree T on n taxa, with internal nodes ordered by their time from the extant taxa. We will then parametrize the tree by mapping it to its ranked topology and a n-1-dimensional vector $\overline{\tau}=(\tau_1,...,\tau_{n-1})$, where$\tau_i$is the time difference between the i-th and i+1-th nodes. The picture below taken from Gavryushkin and Drummond, 2016 represents this parametrization for a tree on 5 taxa. Hence, formally we have the mapping given by $p(T)=(\mathrm{rt}(T), \overline{\tau}(T)$ that embeds the space of ultrametric phylogenetic trees into a disjoint union of $m=\frac{n!(n-1)!}{2^{n-1}}$ (Semple and Steel, 2003) non-negative real n-1-dimensional orthants (i.e. sets of the form $\mathbb{R}^{n-1}_0=\{(x_1,...,x_{n-1}\in\mathbb{R}^n|x_i\geq 0\}$). We will impose an upper bound on all orthants to turn our construction into a cubical complex for the ease of the exposition. However, the construction of 𝛕-space with orthants will still yield a CAT(0) space with the key properties and desiderata (D1)-(D5′) preserved. Note that the ranked topology in this case is a stricter condition on the shape of the tree than just the topology constraint. In particular, the two trees pictured below have distinct ranked topologies, and hence will belong to two distinct orthants in the 𝛕-space. Finally, to properly turn this into a cubical complex we will need to identify the isometries that glue the faces of our cubes together. However, this is rather obvious by construction, since whenever any 𝛕 coordinate becomes 0 we observe a collapse of two nodes in the ranked hierarchy to the same level. Hence, for example the two trees shown above will belong to the cubes that share the $\tau_2=0$ face. Since the resulting space is a cubical complex, we can naturally consider the Euclidean metric within each cube with paths for joining points in different cubes being the sums of the respective within cube paths. The clip below shows how a tree can vary in 𝛕-space while being restricted to a single orthant determined by its ranked topology. Note that just as we discussed above in order to stay within the orthant in 𝛕-space we must keep the ranked topology constant. For comparison, we show how a similar ultrametric tree can vary within a single fixed orthant of a BHV-space. In this case, the topology of the tree is fixed, but the ranked topology can vary. ## Properties After defining the 𝛕-space it is natural to ask whether it manages to achieve our desiderata. In order to check these properties we need to first figure out how many common faces two cubes in our complex can share (recall from the previous post that a theorem of M. Gromov gives characterization of CAT(0) cubical complexes in terms of face-sharing properties). In general, similarly to the previous post by face of a cube we mean any sub-cube, thus it is useful to have a separate term for faces of codimension 1 (i.e. faces that are 1 less dimensional than the cubes). Hence, we will call faces of codimension 1 facets. In particular, any facet can be shared by 1, 2 or 3 cubes in total. If we let $t_1=0$ , then the resulting facet is only contained in the cube of corresponding rooted topology. If setting $t_i=0$ does not result in a multifurcating topology then the corresponding facet will be shared by exactly two cubes (see example in Figure 2 and/or Figure 2 of Gavryushkin and Drummond, 2016). Finally, if we get a multifurcation then the total number of cubes sharing the facet is 3. Now, we are ready to take a stab at the critical result about the 𝛕-space, namely that it is a CAT(0) space, and hence uniquely geodesic. We will recall the theorem of M. Gromov here to reiterate the key result we need to prove. Theorem 1. A cubical complex $K$ with the intrinsic Euclidean metric is CAT(0) if and only if $K$ is connected, simply connected, and for all natural $k$: if three $(k+2)$-cubes of $K$ share a common $k$-cube and pairwise share common distinct $(k+1)$-cubes, then they are contained within a $(k+3)$-cube in $K$. We start by noting that the metric we introduced on our cubical complex is precisely the intrinsic Euclidean metric, and it is easy to see from our construction that the resulting complex is connected and simply connected. To finalize the proof we need to introduce the concept of the link of a vertex v in a complex. We say that the graph G with nodes given by facets containing v and edges given by cubes that contain the two incident nodes as facets is the link of the vertex v. First, we need to show that the cubes of the dimension (k+2) in the theorem cannot be the highest dimensional cubes in the complex. If that was the case, then the link of origin would have to contain a 3-cycle (given by the pairwise distinct (k+1)-cubes) which contradicts the result that the nearest-neighbor interchange graph does not contain 3-cycles. Thus, it follows that each of the (k+2)-cubes has at least one coordinate equal to 0. For the sake of clarity of the presentation we will assume that this is unique coordinate for each of the cubes, although the similar argument can be made in the general case. Let i, j, and r denote the respective coordinates in three cubes. We have three cases to analyze: (i=j=r) This is impossible due to no 3-cycle property of the link of the origin. (i=j) In this case the first two cubes must share a (k+1)-cube, and since all shared (k+1)-cubes must be pairwise distinct it cannot be the cube obtained via setting $\tau_r=0$. Hence, there exists $\tau_s$ s.t. it is greater than zero in both of the first two cubes, and is zero in their shared (k+1)-cube. Now, if the first and third cube share a (k+1)-cube then it follows that their i and r coordinates both have to be zero in the shared cube, implying that the s coordinate has to be resolved in the same way between the first and third cube. However, the same exact argument can be repeated for the second and third cube implying that the first and second cubes had to be indetical. (all distinct) In this case the left out coordinate (r for the first and second cube, i for the second and third, j for the first and third) has to be resolved the same way in the two cubes under consideration. Now, we can construct a (k+3)-cube that contains all three cubes by taking the first cube and resolving its zero coordinate (i) the same way as it is resolved in the remaining two cubes. This analysis concludes the proof of the property required to claim that the cubical complex we defined is indeed CAT(0). It immediately follows that the geodesics in 𝛕-space are unique. At this point we are essentially done with (D1)-(D4) and the only remaining piece is the (efficient) computability of the said geodesics. ## Efficiently computing CAT(0) geodesics In their manuscript Gavryushkin and Drummond, 2016 indicate that the algorithm proposed by Owen and Provan, 2010 will work in 𝛕-space, and provide a link to a Java implementation hosted here: Due to the constraints on the length and scope of the post we will not dive into how does the algorithm of Owen and Provan, 2010 work in the 𝛕-space. Instead, we will discuss the algorithm in its own dedicated post. ## Data and code availability Code used to generate the 𝛕-space and BHV-space tree examples, as well as HTML outputs from Bokeh are provided in the following archive. ## What’s next We spent a noticeable amount of time discussing mathematical properties of different metrics that can be imposed on tree spaces. As our next step, we will switch the lens to a computational perspective and explore the algorithm proposed by Owen and Provan, 2010. We might also touch upon more recent work in the area of the efficient computation of geodesics in CAT(0) cubical complexes. Once we are satisfied with our algorithmic solutions, we will continue the journey by exploring the BHV-space and t-space of phylogenetic trees. ## References Billera, Louis J., Susan P. Holmes, and Karen Vogtmann. “Geometry of the space of phylogenetic trees.” Advances in Applied Mathematics 27, no. 4 (2001): 733-767. https://doi.org/10.1006/aama.2001.0759 Bridson, Martin R., and André Haefliger. Metric spaces of non-positive curvature. Vol. 319. Springer Science & Business Media, 2013. https://doi.org/10.1007/978-3-662-12494-9 Gavryushkin, Alex, and Alexei J. Drummond. “The space of ultrametric phylogenetic trees.” Journal of theoretical biology 403 (2016): 197-208. arXiv:1410.3544 Owen, Megan, and J. Scott Provan. “A fast algorithm for computing geodesic distances in tree space.” IEEE/ACM Transactions on Computational Biology and Bioinformatics 8, no. 1 (2010): 2-13. https://doi.org/10.1109/TCBB.2010.3 Semple, Charles, and Mike Steel. Phylogenetics. Vol. 24. Oxford University Press on Demand, 2003. # Phylogenetic Vignettes: CATs you can’t pet Unlike the previous two posts it will not be obvious how what we will discuss connects to phylogeny. Furthermore, we will be diving into a noticeably more mathematically dense content this time. I will try to use some analogies and visual aids through out this post in order to make the material slightly more accessible. ## Introduction Geometry is a wonderful subject (even if I struggled with it in high school) that makes appearances in many aspects of various sciences (including social ones). However, musing about geometry at large is better left to people who are (a) more proficient in the study of it, and (b) have an acumen for writing more lengthy works, hence I will simply point you in the direction of Jordan Ellenberg’s Shape. What we will concern ourselves with today is a more specific slice of the geometry in which we will look at metric spaces of non-positive curvature. I will not define the notion of a metric space again, as you can simply navigate either to the previous post in the series or Wikipedia. The bulk of this post will consist of defining non-positive curvature, and then we will follow up with a few properties of such spaces, and some examples. Most of the content that follows will adopt the definitions and notation from Bridson and Haefliger, 2013. ## Geodesics Geodesics are a natural notion extending the concept of the “shortest” path curve to general metric space framework. More precisely, given a metric space $(X, d)$ a geodesic from $x$ to $y$ is the map $c:[0, l]\to X$ s.t. $c(0)=x,\, c(l)=y$ and $d(c(t), c(t^\prime))=|t-t^\prime|$. We will call a metric space a geodesic space if for any $x, y\in X$ there exists a geodesic from $x$ to $y$. Furthermore, if for all pairs of points such geodesic is unique we will call such space uniquely geodesic (see Bridson and Haefliger, 2013, pp. 4-8 for additional details and examples). For example, the classic Euclidean space is uniquely geodesic with the geodesics given by straight lines, i.e. for any two points $x,y\in\mathbb{R}^n$ we have the geodesic given by $c(t)=(1-t)x+ty$. On a sphere geodesic between two points $x, y$ is the arc segment obtained by intersecting a plane through $x, y$, and the center of the sphere with the surface (i.e. the great circle arc). An example of a geodesic triangle on a sphere is given below (image courtesy of Wikipedia). Note, that a sphere is not a uniquely geodesic space, since any pair of diametrically opposed points has an infinite set of possible geodesics between them. ## Triangles Just like the notion of a geodesic extends our naïve understanding of the shortest paths, the notion of a geodesic triangle (Bridson and Haefliger, 2013, p. 158) extends our notion of a triangle. Given a metric space $(X, d)$ a geodesic triangle $\Delta$ is a set of tree points $p, q, r\in X$ called vertices and a choice of three geodesic segments $[p, q], [q, r], [r, p]$ joining them called sides, we will denote such a triangle $\Delta([p, q], [q, r], [r, p])$ or more briefly $\Delta(p, q, r)$. Note, that the later notation is not precise, as in the case of a non-uniquely geodesic space there is not necessarily a unique choice of a geodesic between two vertices. Finally, we will write $x\in\Delta$ to indicate that $x$ belongs to the union $[p, q]\cup [q, r]\cup [r, p]$. In order to define a CAT(k) space we also need the notion of a comparison triangle. In order to avoid a lengthy (and somewhat involved) discussion of the nuances about the model spaces and corresponding notation, we will focus on three general types of model spaces: $M_0^n=\mathbb{E}^n$ Euclidean n-dimensional space, $M_{1}^n=\mathbb{S}^n$ the n-sphere, and $M_{-1}^n=\mathbb{H}^n$ hyperbolic n-space. We already defined geodesics in the case of $\mathbb{E}^n$. For the n-sphere we will define the metric via cosine distance $\cos d(x, y) = x^\top y$ where the inner product is in the corresponding embedding $\mathbb{S}^n\to \mathbb{E}^{n+1}$ and the corresponding geodesics are given by minimal great arcs (for full definition and description see Bridson and Haefliger, 2013, pp. 16-17). For the hyperbolic n-space we will use the same construction as Bridson and Haefliger, 2013 (p. 18-20) by considering the space $\mathbb{E^{n, 1}}$ which consists of $\mathbb{R}^{n+1}$ with bilinear form $\langle u|v\rangle = -u_{n+1}v_{n+1}+\sum_{i=1}^n u_iv_i$ and defining $\mathbb{H}^n=\{u\in\mathbb{E^{n, 1}}|\langle u|u\rangle = -1, u_{n+1}> 0\}$. The distance on this space will be given by $\cosh d(x, y) = -\langle x|y\rangle$ (similarly to the sphere case, see Bridson and Haefliger, 2013, pp. 18-23 for details). Finally, a comparison triangle $\overline{\Delta}=\Delta(\overline{p}, \overline{q}, \overline{r})$ is a triangle in the model space $M^2_k$ that satisfies $d(\overline{p}, \overline{q}) = d(p, q)$, $d(\overline{q}, \overline{r}) = d(q, r)$, and $d(\overline{r}, \overline{p}) = d(r, p)$. A point $\overline{x}\in[\overline{p}, \overline{q}]$ for $x\in[p, q]$ is called a comparison point if $d(\overline{p}, \overline{x}) = d(p, x)$. Note, that for $k>0$ we need an additional condition on the perimeter of a triangle to guarantee existence of a comparison triangle, for the purposes of this post we will state the condition, but not elaborate on it in detail. ## CAT(k) So what’s a cat? It’s a gorgeous animal that many humans have as a pet. Many cats are fluffy, and so is mine. Also cats are silly and generally cool to have around. However, this post is not about these kinds of cats. The term “CAT(k)” space was coined by Mikhail Gromov and consists of three initials honoring: Élie Cartan, Alexander D. Alexandrov, and Victor A. Toponogov. Now, we will define a CAT(k) space. Let $X$ be a metric space and let $k$ be a real number. Let $\Delta$ be a geodesic triangle in $X$ with perimeter less than $2D_k$ (the diameter of the $M^2_k$ space, this is the condition need to guarantee the existence of comparison triangle). Then we say that $Delta$ satisfies the CAT(k) inequality if for the corresponding comparison triangle $\overline{\Delta}$ and all $x, y\in\Delta$ with corresponding comparison points $\overline{x}, \overline{y}\in\overline{\Delta}$ the inequality $d(x, y)\leq d(\overline{x}, overline{y})$ is satisfied. Thus, for a $k\leq 0$ we will call $X$ a CAT(k) space if $X$ is geodesic and all its geodesic triangles satisfy the CAT(k) inequality. For $k>0$, we relax definition by only requiring $X$ to be $D_k$-geodesic and only requiring the inequality condition for triangles of perimeter bounded by $2D_k$. Finally, we will call a space to be of curvature at most $k$ if it is locally a CAT(k) space. Hence, any space that is (locally) CAT(0) is a space on non-positive curvature (in the Alexandrov sense). Intuitively speaking the triangles in a CAT(k) space have to be thinner than the ones in the corresponding model space. In particular, triangles in CAT(0) spaces are thinner than those in the regular Euclidean space (namely given two points on the sides of a triangle the distance between them in CAT(0) space is less than or equal to the distance between the corresponding comparison points in the Euclidean space). The illustration below shows three model spaces (sphere, plane, and hyperbolical paraboloid) and a cat drawn in each space. You can see that sphere cat is chubbier than the flat cat, which is chubbier than the hyperbolic cat, turns out triangles in these spaces are also of different chubbiness. For the rest of the post we will focus specifically on the case of the CAT(0) spaces, as they present the most interest to us in the follow up posts. An immediate, but important consequence of the definition of a CAT(k) space is that any CAT(0) space is uniquely geodesic. Note, that being uniquely geodesic also implies that a space is contractible (the converse does not hold, there are contractible spaces that are not uniquely geodesic, a simple example is the closed upper hemisphere which is clearly contractible, and also not uniquely geodesic for the same reason a sphere isn’t). Next, we will briefly describe some interesting spaces which under certain conditions turn out to be CAT(0). ## My CAT is a cubical complex! Let $I=[0, 1]$ be the unit interval, then we call the n-fold product $I^n$ the unit cube. We will let $I^0$ denote a point by convention. Since we will be mainly operating in $n\geq 3$ dimensions, we will use the term “face” to describe any-dimensional face of the cube. Thus, the faces of $I=[0, 1]$ are$\{0\}, \{1\}\$ (the 0-dimensional faces), and $[0, 1]$ (the 1-dimensional face). Analogously, for $I^n$ a face is a subset $S\subseteq I^n$ which can be written as $\prod_{i=1}^n S_i$ where each $S_i$ is a face of $I$. The dimension of the face will be the sum of dimensions of $S_i$, and we also note that a k-dimensional face will be isometric to $I^k$.

Now, we are ready to define a cubical complex (Def. 7.32 from Bridson and Haefliger, 2013) in a similar vein to the definition of a simplicial complex (for those familiar with the latter construction). A cubical complex $K$ is the quotient of a disjoint union of cubes $X=\amalg_\Lambda I^{n_\lambda}$ by an equivalence relation $\sim$ with the restrictions $p_{\lambda}: I^{n_\lambda}\to K$ of the natural projection $p:X\to K$ satisfying

1. for every $\lambda\in\Lambda$ the map $p_\lambda$ is injective;
2. if $p_{\lambda}(I^{n_\lambda})\cap p_{\lambda^\prime}(I^{n_{\lambda^\prime}})\neq\emptyset$ then there is an isometry $h_{\lambda, \lambda^\prime}$ from a face $T_\lambda\subseteq I^{n_\lambda}$ onto a face $T_{\lambda^\prime}\subseteq I^{n_{\lambda^\prime}}$ such that $p_\lambda (x)=p_{\lambda^\prime}(x^\prime)\iff x^\prime=h_{\lambda, {\lambda^\prime}}(x)$.

Note, that in general a cubical complex needs not be a CAT(0) space. However, the following theorem of Gromov, 1987 gives the necessary and sufficient conditions for a cubical complex $K$ to be CAT(0).

Theorem 1. A cubical complex $K$ with the intrinsic Euclidean metric is CAT(0) if and only if $K$ is connected, simply connected, and for all natural $k$: if three $(k+2)$-cubes of $K$ share a common $k$-cube and pairwise share common distinct $(k+1)$-cubes, then they are contained within a $(k+3)$-cube in $K$.

## A bit of perspective

In many cases the underlying geometry of the solution space can make or break our ability to solve problems efficiently. For example, many interesting discrete problems are NP-hard, and often do not even have good approximations (for a deeper dive Google “inapproximability results”, Khot, 2010 gives a great overview). In general, computing geodesics is a hard problem, but in certain spaces we can still efficiently compute (meaning in polynomial time) or approximate geodesics. CAT(0) cubical complexes are one particular class of “nice” spaces, meaning that geodesics can be computed or approximated well enough in polynomial time (Owen, and Provan, 2010, Hayashi, 2021). Implications of these algorithmic results mean that in certain phylogenetic (and robotics) problems we are able to efficiently compute distance between two points in the space, and reconstruct the optimal path between them (or at least do so up to $\varepsilon$ error).

## Code and data availability

All code used in this post is available in the following Jupyter notebook.

## What’s next

The next post in the series will take us back to the world of phylogeny, but this time armed with the knowledge about CAT(0) spaces. We will try to make sense of defining nice metrics on the space of phylogenetic trees with branch lengths, and learn a thing or two in the process. Essentially, the idea for the remainder of this stretch of the series will be to work our way through the paper of Gavryushkin and Drummond, 2016 on the space of ultrametric phylogenetic trees.

## References

Bridson, Martin R., and André Haefliger. Metric spaces of non-positive curvature. Vol. 319. Springer Science & Business Media, 2013. https://doi.org/10.1007/978-3-662-12494-9

Gavryushkin, Alex, and Alexei J. Drummond. “The space of ultrametric phylogenetic trees.” Journal of theoretical biology 403 (2016): 197-208. arXiv:1410.3544

Gromov, Mikhael. “Hyperbolic groups.” In Essays in group theory, pp. 75-263. Springer, New York, NY, 1987. https://doi.org/10.1007/978-1-4613-9586-7_3

Hayashi, Koyo. “A polynomial time algorithm to compute geodesics in CAT (0) cubical complexes.” Discrete & Computational Geometry 65, no. 3 (2021): 636-654. arXiv:1710.09932

Khot, Subhash. “Inapproximability of NP-complete problems, discrete Fourier analysis, and geometry.” In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures, pp. 2676-2697. 2010. https://cs.nyu.edu/~khot/papers/icm-khot.pdf

Owen, Megan, and J. Scott Provan. “A fast algorithm for computing geodesic distances in tree space.” IEEE/ACM Transactions on Computational Biology and Bioinformatics 8, no. 1 (2010): 2-13. https://doi.org/10.1109/TCBB.2010.3

# Phylogenetic Vignettes: Robinson-Foulds metric

Often in biology applications we are interested not only in inferring the optimal tree under some evolutionary model, but also in comparing two different trees for the same set of taxa (or even two different trees for two overlapping but not necessarily identical sets of taxa), it is natural to ask a question about the the dissimilarity measures between trees. In general, the question of defining a metric over the space of phylogenetic trees is quite rich and interesting, and we will explore this in more detail over several posts. However, it is useful to start with the simplest case in which we compare two unrooted trees without branch lengths over a fixed set of taxa. While multiple methods have been proposed to address this question, the metric proposed by Robinson and Foulds, 1981 remains a popular choice in practice (the paper has more than 2.6k citations according to Google Scholar), and more importantly for us serves as a great starting point for the exploration of the metrics on tree spaces.

Before we dive into the details of Robinson-Foulds metric it might be useful to briefly review some of the general metric space concepts and definitions.

## Review: Metric spaces

A metric space is an ordered pair $(X, d)$ where $X$ is a set and $d$ is a function $d: X\times X\to \mathbb{R}$ satisfying

• $d(x, y) = 0 \iff x=y$
• $d(x, y) = d(y, x)$
• $d(x, y) \leq d(x, z) + d(y, z)$

for any $x, y, z\in X$. Recall from the previous post that in the case when the third condition (called triangle inequality) is replaced by a stricter condition $d(x, y)\leq\mathrm{max}\{d(x, z), d(y, z)\}$ we obtain an ultrametric space.

Note that given any set $X$ we can create a simple metric on it by setting $d(x, y) = 1$ if $x\neq y$ and $0$ otherwise. This metric is called the discrete metric (and it is trivial to check that discrete metric is also an ultrametric). Discrete metric provides us with an important realization that any set can be turned into a metric space. However, it is clear that discrete metric is not useful in a practical comparison scenario.

We also note that a metric on a set gives rise to topology, so any metric space is also a topological space. The base for topology on a metric space $X$ is defined by the open balls $B(x, r)=\{y\in X|d(y, x).

In case when the underlying set $X$ is finite, we call the resulting metric space a finite metric space. Note that if we consider trees without branch lengths then the resulting set is finite, and hence the corresponding metric space is a finite metric space. This is not the case when we consider the branch lengths, but we will leave the discussion of the geometry of phylogenetic trees with branch lengths to later posts.

## Robinson-Foulds metric

Given a set of taxa $\mathcal{X}$ let us define the set of all unrooted (not necessarily bifurcating) trees with the property that all nodes of degree 1 or 2 are labeled by pairwise disjoint non-empty subsets of $\mathcal{X}$ as $\mathcal{T}$. Below are a few examples of trees in $\mathcal{T}$ for the set of taxa $\mathcal{X}=\{A, B, C, D, E\}$. Unlabeled nodes are considered to have $\emptyset$ as their label.

Conversely neither of the trees shown below are in $\mathcal{T}$, since they violate the labeling properties.

Note, that any unrooted phylogenetic tree on $\mathcal{X}$ will be an element of $\mathcal{T}$, but not all trees in $\mathcal{T}$ are fully resolved unrooted phylogenies.

Now, we will define two operations on $\mathcal{T}$ named contraction (denoted by $\alpha$) and decontraction (denoted $\alpha^{-1}$).

Contraction will consist of an edge contraction together with the union operation on the corresponding node label sets. More precisely given a tree $T\in\mathcal{T}$ with the labeling defined by $L:V(T)\to\mathcal{P}(\mathcal{X})$ and an edge $\{u, v\}=e\in T$ we will define the contraction $\alpha(T, e)=T^\prime$ by considering $T^{\prime\prime}=(V\setminus\{u, v\}, E(T\setminus\{f\in E(T)|f\text{ is incident to }u{ or }v\}$ and adding a vertex $w$ with $L(w) = L(u)\cup L(v)$ and edges $E_w = \{\{t, w\}|\forall w\text{ s.t. }\{w, u\}\in E(T)\lor\{w, v\}\in E(T)\}$. Thus, we have $\alpha(T, e) = T^{\prime} = (V(T^{\prime\prime})\cup{w}, E(T^{\prime\prime})\cup{E_w})$. The figure below illustrates an example of subsequent contractions on tree in $\mathcal{T}$ for $\mathcal{X}=\{A, B, C, D, E\}$.

Decontraction will consist of an “inverse” operation in which we will split a node by creating an edge. Note that the caveat here is that such an operation depends not only on the choice of the node to split, but also on the choice of a label set and incident edges set split. Hence, the use of notation $\alpha^{-1}$ is misleading since applying a decontraction after a contraction does not necessarily result in the original tree. Hence, to define a decontraction precisely we will define it as a function $\alpha^{-1}(T, v, s, \epsilon) = T^{\prime}$. Let $s = (S_1, S_2)$ where $S_1\cup S_2=L(v)$ and $S_1\cap S_2=\emptyset$. Let $\epsilon = (E_1, E_2)$ where $E_1\cup E_2=\{e\in E(T)|v\in e\}$ and $E_1\cap E_2=\emptyset$. Now, we can construct the tree $T^{\prime\prime}=(V(T)\setminus\{v\}, E(T)\setminus(E_1\cup E_2))$ and add two vertices $u, w$ to it alongside the edge $\{u, w\}$ and redistribute the edges previously incident on $v$. Formally, $\alpha^{-1}(T, v, s, \epsilon) = T^{\prime} = (V(T^{\prime\prime})\cup\{u, w\}, E(T^{\prime\prime})\cup\{\{u, t\}|\exists e\in E_1, t\in e\land t\neq v\}\cup\{\{w, t\}|\exists e\in E_2, t\in e\land t\neq v\})$ with the labeling function defined as $L(u) = S_1$ and $L(w)=S_2$ for the new nodes. The figure below illustrates two of the possible ways a decontraction can be applied to tree in $\mathcal{T}$.

Now, we can finally define the Robinson-Foulds metric on the space of trees $\mathcal{T}$. We let $d(T_1, T_2)$ denote the minimum number of applications of the $\alpha$ and $\alpha^{-1}$ operations required to transform $T_1$ into $T_2$ (whenever we say that two trees are the same we mean that there exists a label preserving isomorphism).

First, let’s make sure that the distance function is well-defined. We note that any $\alpha$ operation can be reversed via an appropriately chosen $\alpha^{-1}$ operation. Furthermore, it is clear that for any tree $T = (V, E)$ within the $|E|$ $\alpha$ operations we will obtain a tree consisting of a single node with the label $\mathcal{X}$ (representing all taxa) and no edges. Thus, for any two trees $T_1, T_2$ it follows that we can transform both to the tree with one node, and then apply appropriate inverse transformations, implying that there is a finite (and in fact bounded above by $2|\mathcal{X}|-2$) sequence of $\alpha, \alpha^{-1}$ operations that transforms $T_1$ into $T_2$ (and vice versa). We will not prove that $d(T_1, T_2)$ is indeed a metric, but the proofs of each of the properties are easy.

## Robinson-Foulds metric: properties and caveats

Robinson-Foulds (RF) metric comes with several nice properties which made it ubiquitous in the world of phylogenetic analyses: (1) it can be efficiently computed (linear time in number of taxa via Day, 1985 or approximate solution in sub-linear time), and (2) it has intuitive interpretation as the number of distinct splits between the two trees (for more details see Felsenstein, 2004). However, RF metric also suffers from several issues. First, the placement of a single taxon can easily produce maximum RF distance between two otherwise identical trees. The figure below illustrates this example for two unrooted trees with 5 taxa.

Furthermore, the problem of large RF distances is a general issue with this metric. For example, we have generated 945 rooted trees on 6 taxa, and computed pairwise distance between distinct pairs of trees. The resulting histogram of normalized RF distances is shown below.

As we can see more than a third of all distances achieved the maximum possible value. Put differently, if we view the discrete metric as the least informative (everything is distance 1 away from any other point) then RF metric is only slightly better than the discrete one. This obviously presents us with a problem when we want our comparisons to provide a relative ranking of similarity with respect to a reference tree or when we want to use the distances as a tool for comparison between multiple sets of data.

There are a number of refinements of the RF metric which attempt to overcome some of the issues we pointed out above. Of particular interest is recent work of Smith, 2020 in which the author proposes an information theoretic generalization of the RF metric. Key idea in this paper is to use the information content of each split in order to weight its impact on the measured distance. Furthermore, the paper provides a set of comparisons with other generalizations of the RF metric, and as such can serve as a good overview of the topic for readers curious to learn more about the current state of the field.

## Data and code availability

All code used in this post is available as a Jupyter notebook in the attached archive.

## What’s next?

Robinson-Foulds metric serves as an important example at the start of the journey into tree geometry and comparative analyses. However, since it is purely focused on the topology of the tree it lacks expressiveness required to compare phylogenetic trees with branch lengths. Thus, we need different methods in order to compare those. Furthermore, the arising geometries (spoiler alert: there is more than one) are quite interesting in themselves. Thus, we will likely return to the topic of tree distances later on in the series.

## References

Day, William HE. “Optimal algorithms for comparing trees with labeled leaves.” Journal of classification 2, no. 1 (1985): 7-28. https://doi.org/10.1007/BF01908061

Felsenstein, Joseph. Inferring phylogenies. Vol. 2. Sunderland, MA: Sinauer associates, 2004.

Robinson, David F., and Leslie R. Foulds. “Comparison of phylogenetic trees.” Mathematical biosciences 53, no. 1-2 (1981): 131-147. https://doi.org/10.1016/0025-5564(81)90043-2

Smith, Martin R. “Information theoretic generalized Robinson–Foulds metrics for comparing phylogenetic trees.” Bioinformatics 36, no. 20 (2020): 5007-5013. https://doi.org/10.1093/bioinformatics/btaa614

# Phylogenetic Vignettes: UPGMA and Neighbor-joining

In an effort to learn more about phylogenetics and practice my writing skills I am starting a mini-series of phylogenetic vignettes in which I will explore different concepts in phylogenetic analysis and their implementations. This series is intended as a partial account of my journey in learning some of the topics in phylogenetics. The resulting posts are not intended to be complete explorations of the corresponding questions, and will be presented at the current level of author’s understanding which can be incomplete. The intended audience for these vignettes are undergraduate and graduate students with solid understanding of basic concepts in discrete mathematics, as well as some knowledge of analysis and probability. I will assume no deep knowledge of evolutionary biology, in part due to my own background lacking strong biology foundation.

## Preliminaries

We will follow the notation adopted in Huson, Rupp, and Scornavacca, 2010.

We will also use the term molecular clock to refer to the situations in which the rate of mutation is fixed per unit of time. Thus the molecular clock hypothesis is a useful assumption for inference of trees for the sequences that were collected at the same point in time, hence we can assume in that scenario that in the rooted phylogenetic tree distance from root $\rho$ to every leaf is constant.

## Distance based tree inference

In general, phylogenetic tree inference problem consists of inferring an optimal rooted or unrooted phylogenetic tree $T$ given a set of sequences representing taxa $\mathcal{X}$. Here by optimal we mean optimization with respect to some evolutionary model over the full parameter space defined by topologies of $T$ and weights $\omega$.

The exact criterion with respect to which the optimization is done varies between various formulations of the problem. However, one thing that is common to all formulations is the prohibitively large size of the search space. A simple calculation shows that the number of bifurcating trees (i.e. every non-leaf node has exactly two children or alternatively every non-leaf has degree 3) on $n$ taxa is $(2n-3)!!$.

In the case of distance based phylogenetic tree inference our input is the $n\times n$ pairwise distance matrix $D$. We will explore different options for measuring distance between input sequences in a later section.

## Least squares

We can start with an attempt at defining an optimization goal for our tree inference problem given the input distance matrix $D$. Intuitively the goal here is to find a tree that best approximates (explains) the observed distances. More formally, given a phylogenetic tree $T$ with edge weights $\omega$ we can define the tree-induced metric on the set of taxa $\mathcal{X}$ as $d_T(x, y)=\sum_{e\in x\sim y}\omega(e)$, where we use the notation $x\sim y$ to denote the path between leaves $x, y$ of the tree (in a tree any two nodes are connected by a single path). Thus, we want to minimize the error $\parallel D-D_T\parallel$ under some norm. One possible choice of the norm is the Frobenius norm, which in turn corresponds to unweighted least squares tree inference. As noted in Felsenstein, 2004 the least squares formulation can be weighted, i.e. the function to minimize becomes $\sum_{i=1}^n\sum_{j=1}^nw_{ij}(D_{ij}-{D_T}_{ij})^2$ with the choice $w_{ij}=1$ proposed by Cavalli-Sforza and Edwards, 1967, $w_{ij}=1/D^2_{ij}$ by Fitch and Margoliash, 1967, and $w_{ij}=1/D_{ij}$ by Beyer et al., 1974.

While for a fixed tree topology the problem boils down to least squares optimization, the general problem of tree inference under this model requires searching over the large space of trees. Thus, while the formulation behind this method is sound and appealing, in practice we tend to rely on heuristics when searching the tree space.

Instead of fully exploring least squares method and its derivatives and search heuristics we will now pivot to a different approach in which instead of explicit optimization under a given metric we will instead focus on direct clustering goal.

## UPGMA

The first approach we will explore is called Unweighted Pair Group Method with Arithmetic Mean (UPGMA). For those familiar with clustering methods, and hierarchical clustering in particular, this name might ring a bell. It appears that the name UPGMA mostly stuck with the phylogenetics community, and in the general clustering world this approach is referred to as agglomerative (i.e. bottom-up) clustering with average linkage (scikit-learn reference, scipy reference).

Implementing UPGMA naively is straightforward and we will illustrate it in the following code snippet with the distance matrix data coming from Felsenstein, 2004 (pp. 162-166).

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Note that the above implementation doesn’t keep track of the branch lengths when providing the output and as a result our tree has the correct topology but misses the edge weight information. Alternatively we can simply use Scipy hierarchical clustering module and achieve the same result including the branch lengths in fewer lines of code.

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One thing that becomes apparent from the second plot is that the resulting tree is an ultrametric tree.

In general we call a metric space $(M, d)$ and ultrametric space if in addition to the standard properties the metric $d$ satisfies the strong triangle inequality $d(x, y)\leq\mathrm{max}\{d(x, z), d(y, z)\}$ for all $x, y, z\in M$. Ultrametric spaces are somewhat bizarre looking since every triangle in such space is isosceles. Furthermore, every point inside an open ball of positive radius is its center, and all open balls of positive radius are clopen. However, ultrametric spaces arise naturally in several settings with the one of interest to us being phylogenetic trees under the molecular clock hypothesis. Such trees are commonly referred to as ultrametric trees.

Now, one of the interesting properties of the UPGMA algorithm is that the tree constructed by this method will always be ultrametric. However, this also implies that if the original distance matrix does not induce an ultrametric on $\mathcal{X}$ then the tree produced by the UPGMA algorithm will have a non-zero error $\parallel D-D_T\parallel$. For example, consider the following sample distance matrix given as an example in Huson, Rupp, and Scornavacca, 2010 (p. 53):

The inferred UPGMA tree will look as following

Due to the metric induced by the tree the distances between C and D, and C and E, will be equal in $T$ while we know from the original distance matrix that the distance from C to D is supposed to be larger. Thus, UPGMA model is not broadly applicable since it does require the original distance matrix to satisfy the ultrametric condition. Conversely, Huson, Rupp, and Scornavacca, 2010 Lemma 3.13.2 shows that if the distance matrix $D$ is ultrametric then the tree inferred by UPGMA satisfies $D_T=D$.

In terms of the computational complexity UPGMA requires $O(|\mathcal{X}|^3|)$ time, which is a significant speed up when compared to brute-force search over the tree space.

Felsenstein, 2004 notes that in some scenarios, for example for closely related species, where the molecular clock is a reasonable starting assumption the method beahves well. However, in the general setting where we cannot claim a fixed rate for the whole tree UPGMA approach becomes problematic.

## Neighbor-joining

Neighbor-joining is an alternative method for inferring a phylogenetic tree from the distance matrix $D$. Originally it was proposed by Saitou and Nei, 1987 and has since been implemented in several libraries and software packages. We will use the implementation provided in scikit-bio (in case if you get a conflict with Scipy, try downgrading Scipy to version 1.5.2), and skip discussing the details of the algorithm in this post.

Motivation for considering neighbor-joining is that it doesn’t require the underlying distance matrix to be ultrametric in order to faithfully infer a phylogenetic tree. However, in order to guarantee that the algorithm will infer a tree $T$ with $D_T=D$ we still require that the distance matrix is “tree-like”. This vague term can be made precise via the Definition 3.12.4 from Huson, Rupp, and Scornavacca, 2010 (p. 51) also referred to as the four-point condition: a distance matrix $D$ is said to satisfy the four-point condition if for every quadruple of taxa $x,y,z,w\in\mathcal{X}$ we have $d(w, x)+d(y, z)\leq \mathrm{max}\{d(w,y)+d(x,z), d(w,z)+d(x,y)\}$. In the case when the distance matrix $D$ satisfies the above condition we can guarantee that $D_T=D$ for the inferred tree $T$. The proof of the above claim can be found in the original paper.

Neighbor-joining has the same computational complexity as UPGMA, and both methods are agglomerative clustering approaches that differ in the distance and update rules. One interesting aspect of the neighbor joining method is that it has been shown to greedily optimize balanced minimum evolution metric on the tree space (Gascuel and Steel, 2006). We will not explore the details of the balanced minimum evolution model at this point, but we might return to the topic in later post.

## Exploring sequence distance metrics

So far we only considered examples in which we had a predefined distance matrix $D$ provided to us from the beginning. In this section we will explore a few choices for constructing $D$ given a set of genomic sequences and compare how different distance based inference methods fair in these scenarios.

### Jukes-Cantor model and distance

The simplest substitution model used in phylogenetics is the Jukes-Cantor model. Under this model we assume that the substitution rate is fixed and equal and the base frequencies are all equal to 1/4. Under these assumptions it is easy to derive the correspondence between the expected number of substitutions (i.e. evolutionary distance) and the fraction of distinct bases in the alignment (i.e. normalized Hamming distance). Thus, given two aligned sequences $x, y$ we will refer to the quantity $JC(x, y)=-\frac{3}{4}\ln\left(1-\frac{4}{3}H(x, y)\right)$, where $H(x, y)$ is the normalized Hamming distance, as the Jukes-Cantor distance between $x$ and $y$.

### Mash distance

MinHash sketching is an effective computational technique for estimating similarity of two sets of data. Due to its computational efficiency MinHash has found its into many data intensive areas of computer science, including bioinformatics. In particular, due to the computational efficiency it’s common that we view sequences as sets (or multi-sets) of k-mers (substrings of length k, think k-grams in NLP), and hence we can compare the (Jaccard) similarity of two sequences via MinHash on the respective k-mers. However, upon closer inspection it is clear that Jaccard distance is not a good proxy for the evolutionary distance (similarly how normalized Hamming distance is not exactly the same thing as Jukes-Cantor distance). Thus, we want to compute an appropriate adjustment that turns our estimated Jaccard index into an approximate evolutionary metric. One possible approach is given by Ondov et al., 2006, Eq. 4 which defines Mash distance between two sequences (or sets of sequences) as $D=-\frac{1}{k}\ln\frac{2j}{1+j}$ where $k$ is the length of the k-mer and $j$ is the estimated Jaccard index.

### Comparing UPGMA and Neighbor-joining for JC distance

We will begin with a set of 5 sequences each consisting of 70 nucleotides and containing 5 single nucleotide changes with respect to the first 70 nucleotides of the N gene of the reference genome for SARS-CoV-2.

Reference:
ATGTCTGATAATGGACCCCAAAATCAGCGAAATGCACCCCGCATTACGTTTGGTGGACCCTCAGATTCAA
Sequences:
ATTGCTGATACTGGACCCCAAAATCAGCGAAATGCACCCCGCATTACGTTTGGTGGAACCTCAGATCCAA
ATGTTTGATAATGGACCCCAAATTCAGCGAAATGCACGCCGCATTACGTTCGGTGGACCCTCAGATTCCA
ATGTTTGATAATGGACCCCAAATTCAGCGAAATGCACCCTGCATTACGTTCGGTGGACCCTCAGATTCGA
ATGTCCGATAATGGACCCCAAATTCAGCGAAATGCACTCCGCATTACGTTTGATGGACCCTCAGATTCCA
ATGTCCGATAATGGACCCCAAATTCAGCGAAATGCACCCGGCATTACGTTTGATGGACCCTCAGATCCAA

We will label the sequences in this set A, B, C, D, and E in top to bottom order. We get the following matrix of Jukes-Cantor distances for this set:

It is easy to check that this matrix is not additive, and hence is also not an ultrametric. Thus, we do not expect either of the algorithms to yield an optimal tree. Nevertheless we would like to compare the two topologies we get. First we will consider the UPGMA tree:

We can see that the sequences A ends up acting as an outgroup, which is not surprising if we return and examine the mutation content within the sequence. Sequences B and C, and D and E form two clades in the UPGMA tree. Examining the alignment by eye it appears as a plausible grouping. However, we note that the distance between sequences B and D is estimated to be smaller than that between B and E, while the inferred tree does not capture this difference.

Next, we will consider the tree obtained from the neighbor-joining method.

BCDEA
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Similarly to the UPGMA tree the sequences B and C end up in the same clade in the neighbor-joining tree. Analogously sequence A appears to form an outgroup. The key difference in this case is the way in which sequences D and E are placed onto the tree. However, neighbor-joining tree also fails to capture some of the differences in the distances, for example sequence E is equidistant from B and C in the tree.

### Comparing UPGMA and Neighbor-joining for Mash distance

We will start out with the same sequence setup as in the previous run, except instead of the Jukes-Cantor distances we will use Mash distances this time. We are setting the k-mer size to 7 for this scenario, and we obtain the following distance matrix:

Unsurprisingly this matrix is not additive either, and the distance values tend to correlate with the Jukes-Cantor distances. Thus, as expected both inferred trees have the same topologies as in the previous section, albeit slightly different branch lengths.

BCDEA
(function() { var modules={}; })();

Now, we will try to perform a similar comparison but with a different underlying dataset. We downloaded 454 complete SARS-CoV-2 genomes collected between August 1st, 2022 and August 15th, 2022 from NCBI Virus database. We then computed the Mash distances between all these sequences using 11-, 17-, 23,-, and 31-mers. Finally for each distance matrix we inferred the phylogenetic tree using UPGMA and neighbor-joining methods.

In the above dendrogram we truncated subtrees after a fixed depth level for the ease of visualization, the labels when present indicate a leaf node with the corresponding PANGO lineage assigned to it. One thing that we immediately notice in this case is that BA.2, BA.5 (and BA.4, labels not shown) form a clade, and B.1.617.2, AY.* form a clade. This matches the expected behavior since, the first clade corresponds to the BA.2-related sublineages of Omicron variant, and the second clade corresponds to Delta variant. What is slightly surprising is that BA.1 (also a sublineage of Omicron) ends up in a clade with B.1.616 and B.1 lineages, rather than next to BA.2. We also inferred a neighbor-joining based tree and adopted leaf coloring scheme similar to the one in UPGMA.

(function() { var modules={}; })();

In this tree the deeper shade of orange represents BA.5 and its sublineages, while lighter shade of orange corresponds to BA.2 and BA.4 sublineages of Omicron. Green is used for Delta, and red for everything else. One thing that becomes more apparent with this tree is that there is subclade consisting predominantly of BA.5 sequences that is at a considerable distance from the rest of the sequences in the tree.

Another interesting observation we get from this experiment is that the k-mer size used did not influence the resulting tree shape drastically. While there are some minor changes occurring as we increase the size of k, the main clades centered around BA.5 and Delta emerge in all inferred trees. This suggests a practical observation that in the case of a large amount of closely related sequences, we can apply very approximate methods (since we stacked Mash distance with UPGMA we effectively “approximated” twice) and still capture some underlying structure in the data.

## Data and code availability

In addition to the two Gists shared above you can also find the Jupyter Notebook with the code used to generate figures for this post, as well as sequence data and precomputed Mash distances in the archive attached below.

## What’s next

We very briefly touched upon some of the distance-based phylogenetic inference methods and checked out several trees inferred from two evolutionary distance metrics. However, we haven’t rigorously defined how exactly two tree topologies and two phylogenetic trees can be compared. We will dive into that topic over the course of the next few posts.

## References

Beyer, William A., Myron L. Stein, Temple F. Smith, and Stanislaw M. Ulam. “A molecular sequence metric and evolutionary trees.” Mathematical Biosciences 19, no. 1-2 (1974): 9-25. https://doi.org/10.1016/0025-5564(74)90028-5

Cavalli-Sforza, L.L. and Edwards, A.W.F. (1967), Phylogenetic analysis: Models and estimation procedures. Evolution, 21: 550-570. https://doi.org/10.1111/j.1558-5646.1967.tb03411.x

Felsenstein, Joseph. Inferring phylogenies. Vol. 2. Sunderland, MA: Sinauer associates, 2004.

Fitch, Walter M., and Emanuel Margoliash. “Construction of phylogenetic trees.” Science 155, no. 3760 (1967): 279-284. https://doi.org/10.1126/science.155.3760.279

Gascuel, Olivier, and Mike Steel. “Neighbor-joining revealed.” Molecular biology and evolution 23, no. 11 (2006): 1997-2000. https://doi.org/10.1093/molbev/msl072

Haeckel, E. H. P. A. (1866).Generelle Morphologie der Organismen : allgemeine Grundzüge der organischen Formen-Wissenschaft, mechanisch begründet durch die von C. Darwin reformirte Decendenz-Theorie. Berlin. [Featured image]

Huson, Daniel H., Regula Rupp, and Celine Scornavacca. Phylogenetic networks: concepts, algorithms and applications. Cambridge University Press, 2010.

Ondov, Brian D., Todd J. Treangen, Páll Melsted, Adam B. Mallonee, Nicholas H. Bergman, Sergey Koren, and Adam M. Phillippy. “Mash: fast genome and metagenome distance estimation using MinHash.” Genome biology 17, no. 1 (2016): 1-14. https://doi.org/10.1186/s13059-016-0997-x

Saitou, Naruya, and Masatoshi Nei. “The neighbor-joining method: a new method for reconstructing phylogenetic trees.” Molecular biology and evolution 4, no. 4 (1987): 406-425. https://doi.org/10.1093/oxfordjournals.molbev.a040454
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