Phylogenetic Diversity Rankings in the Face of Extinctions: The Robustness of the Fair Proportion Index

Abstract

Planning for the protection of species often involves difficult choices about which species to prioritize, given constrained resources. One way of prioritizing species is to consider their “evolutionary distinctiveness,” (ED) that is, their relative evolutionary isolation on a phylogenetic tree. Several evolutionary isolation metrics or phylogenetic diversity indices have been introduced in the literature, among them the so-called Fair Proportion (FP) index (also known as the ED score). This index apportions the total diversity of a tree among all leaves, thereby providing a simple prioritization criterion for conservation. Here, we focus on the prioritization order obtained from the FP index and analyze the effects of species extinction on this ranking. More precisely, we analyze the extent to which the ranking order may change when some species go extinct and the FP index is recomputed for the remaining taxa. We show that for each phylogenetic tree, there are edge lengths such that the extinction of one leaf per cherry completely reverses the ranking. Moreover, we show that even if only the lowest-ranked species goes extinct, the ranking order may drastically change. We end by analyzing the effects of these two extinction scenarios (extinction of the lowest-ranked species and extinction of one leaf per cherry) for a collection of empirical and simulated trees. In both cases, we can observe significant changes in the prioritization orders, highlighting the empirical relevance of our theoretical findings. [Biodiversity conservation; Fair Proportion index; phylogenetic diversity; species prioritization.]

Evolutionary isolation measures or phylogenetic diversity indices have become an increasingly popular tool to prioritize species for conservation (e.g., Vane-Wright et al. 1991; Redding and Mooers 2006; Isaac et al. 2007; Redding et al. 2008, 2014; Vellend et al. 2011). These indices assess the importance of species for overall biodiversity based on their placement in an underlying phylogenetic tree and can thus, next to other criteria such as threat status, serve as a prioritization tool in conservation planning. For instance, a species that is only distantly related to others in its family may be prioritized for preservation over one that is more closely related, in order to preserve more breadth in the biodiversity.

One simple index that has been introduced in this regard is the “Fair Proportion (FP) index,” also known as the “evolutionary distinctiveness” (ED) score (Redding 2003; Isaac et al. 2007). The FP index apportions the total diversity of a tree (measured as the sum of edge lengths of the tree) among all leaves by distributing each edge length equally among descending leaves. It is employed in the so-called “EDGE of Existence” project established by the Zoological Society of London, a conservation initiative focusing specifically on threatened species that represent a high amount of unique evolutionary history (Isaac et al. 2007; see also https://www.edgeofexistence.org/). While the FP index itself lacks a direct biological link to conservation, it coincides with the so-called Shapley value, a result that was first shown by Fuchs and Jin (2015). The Shapley value, a concept from cooperative game theory (Shapley 1953), on the other hand reflects the average contribution of a species to the total diversity of a tree and is characterized by certain desirable axioms, namely efficiency, symmetry, additivity, and a “dummy” axiom, that motivate its use (and hence that of the FP index) as a prioritization criterion in biodiversity conservation. Efficiency states that the total diversity of a tree is apportioned among all leaves. Symmetry implies that two species playing the same role in a tree and contributing the same amount of diversity receive the same FP index, while the dummy axiom states that a species not contributing any worth to diversity receives an FP index of zero. Finally, additivity constitutes a technical property regarding “sums of games” that is of less relevance here (for details see, e.g., Haake et al. 2008).

In this article, we focus on the ranking order obtained from the FP index and analyze its robustness to species extinction. More precisely, we consider the following scenario: Suppose you use the FP index to rank species for conservation, that is, to allocate resources to protect some but not all species under consideration. Now suppose one or more species go extinct, for example, because they did not receive any conservation attention. This will result in a change in the underlying phylogenetic tree, because some species are now extinct. If you now recompute the FP index on this revised tree, how confident can you be that your initial choice of priority is unchanged?

Here, we investigate circumstances in which after such an extinction event, the ranking order obtained from the FP index and thus conservation priorities might radically change. We consider scenarios where several species go extinct, as well as scenarios where only one species goes extinct.

The article is organized as follows. We first introduce all relevant concepts and notations. We then show that for each phylogenetic tree, there are edge lengths such that the extinction of one leaf per cherry completely reverses the ranking of the “surviving” species obtained from the FP index (Theorem 2). More precisely, the ranking of the surviving species on the reduced phylogenetic tree and the ranking of the same set of species on the original phylogenetic tree are in reverse order. Afterwards, we analyze cases in which only one species, for example, the lowest-ranked one, goes extinct, and show that this can already have drastic effects on the prioritization order (Theorems 3 and 4). After briefly considering the case of ultrametric caterpillar trees, we complement our theoretical results by studying the effects of species extinction for a collection of empirical trees obtained from the TreeBase database (Piel et al. 2002; Vos et al. 2012), the TimeTree database (Kumar et al. 2017), as well as for simulated data. In both cases, we can observe changes in the prioritization orders when species go extinct, indicating that our theoretical results are not merely mathematical artifacts but are of practical importance. However, these studies also show that the rankings become more “robust” to species extinctions the larger the tree. We end by discussing our results and indicating directions for future research.

Definitions and Background

Phylogenetic |$X$|-Trees and Related Concepts

Let |$X$| denote a nonempty finite set (of taxa) with |$|X|=n$|⁠. A rooted binary phylogenetic |$X$|-tree |$T=(V(T),E(T))$| is a rooted tree (or, more precisely, an arborescence) with root vertex |$\rho$| of in-degree 0 and out-degree 2, where all edges are directed away from the root, all interior vertices apart from |$\rho$| have in-degree 1 and out-degree 2, and the leaves (also referred to as taxa) are bijectively labeled by |$X$|⁠. For technical reasons, if |$|X|=1$|⁠, we additionally allow |$T$| to consist of a single vertex, which is at the same time the root and only leaf of |$T$|⁠. Since all phylogenetic |$X$|-trees in this article are rooted and binary, we will often refer to them simply as phylogenetic |$X$|-trees or trees. Moreover, we call the graph-theoretical tree without leaf labels underlying |$T$|⁠, the tree shape or topology of |$T$|⁠. Furthermore, the edges incident to the leaves are referred to as pendant edges, whereas all other edges are called inner edges. Additionally, we assume that each edge |$e$| of |$T$| is assigned a strictly positive edge length |$\lambda_T(e) \in \mathbb{R}_+$|⁠, representing time or evolutionary distance. Whenever there is no ambiguity we simply refer to the length of an edge |$e$| as |$\lambda_e$|⁠. Moreover, we call |$T$| an ultrametric tree if the path lengths from the root to all leaves of |$T$| are identical. Note that the concept of ultrametric trees is also often referred to as the molecular clock hypothesis in biology.

A vertex |$v$| of |$T$| is a descendant of a vertex |$u$| of |$T$| (and |$u$| is an ancestor of |$v$|⁠), if |$u$| lies on the unique path from |$\rho$| to |$v$| in |$T$|⁠. In particular, a vertex |$u$| is the parent of a vertex |$v$| in |$T$| and |$v$| is a child of |$u$|⁠, if |$v$| is a descendant of |$u$| and the edge |$e=(u,v)$| exists, that is, |$e=(u,v) \in E(T)$|⁠. Then, a cherry |$[x_i,x_j]$| of |$T$| is a pair of taxa |$x_i,x_j \in X$| such that |$x_i$| and |$x_j$| have the same parent in |$T$|⁠. We use |$c_T$| to denote the number of cherries of |$T$|⁠. A phylogenetic tree that has precisely one cherry is called a caterpillar tree (note that up to permuting leaf labels, the caterpillar tree is unique).

When |$n \geq 2$|⁠, we will also often decompose |$T$| into its two maximal pendant subtrees |$T_a$| and |$T_b$| rooted at the children |$a$| and |$b$| of |$\rho$|⁠, and we denote this decomposition by |$T=(T_a,T_b)$|⁠. We use |$X_a$| and |$X_b$| to denote the leaf sets of |$T_a$| and |$T_b$|⁠, respectively. Moreover, we use |$n_a$| and |$n_b$| to refer to |$|X_a|$| and |$|X_b|$|⁠, respectively, and assume without loss of generality that |$n_a \geq n_b \geq 1$|⁠.

Finally, for a subset |$Y\subseteq X$|⁠, the induced subtree |${T_Y}$| of |$T$| is the rooted phylogenetic |$Y$|-tree obtained from the minimal subtree of |$T$| spanning the taxa in |$Y$| by suppressing all nonroot degree-2 vertices, and adding up the edge lengths of edges that are “merged” into a new edge.

The FP Index

The FP index apportions the total sum of edge lengths of |$T$| (also referred to as the “phylogenetic diversity” of |$X$| (Faith 1992) among the taxa in |$X$| (Redding 2003; Isaac et al. 2007). More precisely, the FP index for |$x \in X$| is defined as

$$\begin{equation}\label{Def_FP}{\it FP}_T(x) = \sum\limits_{e \in P(T;\rho,x)} \frac{\lambda_e}{D_e},\end{equation}$$

(1)

where |$P(T; \rho,x)$| denotes the path in |$T$| from the root to leaf |$x$| and |$D_e$| is the number of leaves descended from the head of edge |$e$|⁠. Note that the definition of the FP index does not require |$T$| to be a binary rooted phylogenetic |$X$|-tree and is equally valid for nonbinary phylogenetic trees. Essentially, the FP index distributes each edge length equally among descending leaves. It is thus not hard to see that |$\sum\limits_{x \in X} FP_T(x) = \sum\limits_{e \in E(T)} \lambda_e$|⁠. As an example, for tree |$T$| depicted in Figure 1 and taxon |$x_1$|⁠, we have |$FP_T(x_1)=\frac{1}{6} + 61 = 61 \frac{1}{6}$|⁠.

Figure 1.

A phylogenetic tree |$T$| on |$X=\{x_1, \ldots, x_9\}$|⁠, with edge lengths that induce a strict and reversible ranking when leaves |$x_3$|⁠, |$x_6,$| and |$x_8$| (i.e., one leaf per cherry) are deleted from |$T$| to form |$\widetilde{T}$|⁠. More precisely, |$ {\it FP}_{T}(x_1) > {\it FP}_{T}(x_2) > \ldots > {\it FP}_{T}(x_9) $|⁠, but when leaves |$x_3$|⁠, |$x_6$|⁠, and |$x_8$| are deleted, we have |$ {\it FP}_{\widetilde{T}}(x_1) < {\it FP}_{\widetilde{T}}(x_2) < \ldots < {\it FP}_{\widetilde{T}}(x_9)$|⁠. Thus, the induced ranking |$\pi_{T}$| is strict and reversible.

Strict and Reversible Rankings

Recall that a ranking |$\pi(S,f)$| for a set |$S=\{s_1,\ldots,s_n\}$| based on a function |$f:S\rightarrow \mathbb{R}$| is an ordered list of the elements of |$S$| such that |$f(s_i)\geq f(s_j)$| if and only if |$s_i$| appears before |$s_j$| in |$\pi$|⁠. A ranking function |$f$| is called strict if it is one-to-one, that is, if there are no ties. In other words, |$f$| is strict if |$f(s_i)\neq f(s_j)$| for all |$i\neq j$|⁠. Note that in the following we mostly consider rankings |$\pi(X,FP_T)$|⁠, where |$T$| is a phylogenetic |$X$|-tree. Therefore, whenever there is no ambiguity, we use the shorthand |$\pi_T$| instead of |$\pi(X,FP_T)$|⁠.

We are interested in diversity rankings whose order is reversed if a species or set of species goes extinct. Formalizing this intuition, we call a ranking |$\pi_T$|reversible if there is a subset |$X'$| of |$X$| whose removal from |$X$| and from |$T$| leads to an induced subtree |$\widetilde{T}=T_{\widetilde{X}}$| of |$T$| (with |$\widetilde{X}:=X\setminus X'$|⁠) whose corresponding ranking |$\pi_{\widetilde{T}}=\pi(\widetilde{X}, {\it FP}_{\widetilde{T}})$| ranks the species in the opposite order to |$\pi_T$|⁠. In particular, if |$ {\it FP}_{\widetilde{T}}(x_i)>FP_{\widetilde{T}}(x_j)$| in |$\pi_{\widetilde{T}}$|⁠, then |$ {\it FP}_T(x_i)< {\it FP}_T(x_j)$| in |$\pi_T$|⁠.

An example of a tree |$T$| with edge lengths that induce a strict and reversible ranking is given in Figure 1.

Kendall’s |$\tau$| Rank Correlation Coefficient

The Kendall’s |$\tau$| rank correlation coefficient can be used to quantify the similarity and association of ranked data obtained from different ranking functions. Let |$\pi_1(S,f_1)$| and |$\pi_2(S,f_2)$| be two rankings for a set |$S = \{s_1, \ldots, s_n\}$|⁠. Note that these rankings do not need to be strict but may contain ties. We say that a pair |$(s_i,s_j)$| of elements from |$S$| with |$i < j$| is a concordant pair if |$s_i$| and |$s_j$| are in the same order in |$\pi_1$| and |$\pi_2$| (i.e., |$f_1(s_i) > f_1(s_j)$| and |$f_2(s_i) > f_2(s_j)$|⁠; or |$f_1(s_i) < f_1(s_j)$| and |$f_2(s_i) < f_2(s_j)$|⁠). On the other hand, if |$s_i$| and |$s_j$| are in the opposite order in |$\pi_1$| and |$\pi_2$| (i.e., |$f_1(s_i) > f_1(s_j)$| and |$f_2(s_i) < f_2(s_j)$|⁠; or |$f_1(s_i) < f_1(s_j)$| and |$f_2(s_i) > f_2(s_j)$|⁠), then |$(s_i,s_j)$| is called a discordant pair. Let |$n_c$| denote the number of concordant pairs, let |$n_d$| denote the number of discordant pairs, and let |$n_{\pi_1}$| and |$n_{\pi_2}$| denote the number of pairs that are tied only in |$\pi_1$| or only in |$\pi_2$|⁠, respectively (if a tie occurs for the same pair in both |$\pi_1$| and |$\pi_2$|⁠, it is not added to either |$n_{\pi_1}$| or |$n_{\pi_2}$|⁠). Then, the Kendall’s |$\tau$| coefficient (or Kendall’s |$\tau_b$| as this version of the coefficient, which allows for ties, is often called) is defined as

$$\begin{align*}\tau =\frac{n_c-n_d}{\sqrt{(n_c+n_d+n_{\pi_1})(n_c+n_d+n_{\pi_2})}}.\end{align*}$$

Note that |$\tau \in [-1,1]$|⁠, where the rankings are the same if |$\tau=1$| and they are completely reversed if |$\tau=-1$|⁠. If |$\tau=0$|⁠, the rankings are uncorrelated.

As an example, for tree |$T$| on |$X=\{x_1, \ldots, x_9\}$| depicted in Figure 1 and tree |$\widetilde{T}$| on |$\widetilde{X}=X \setminus \{x_3,x_6,x_8\}$| obtained from |$T$| by deleting leaves |$x_3$|⁠, |$x_6$|⁠, and |$x_8$|⁠, we have |$\pi_T(\widetilde{X}, FP_T) = (x_1, x_2, x_4, x_5, x_7, x_9)$| and |$\pi_{\widetilde{T}}(\widetilde{X}, FP_{\widetilde{T}}) = (x_9, x_7, x_5, x_4, x_2, x_1)$|⁠. As |$|\widetilde{X}|=6$|⁠, there are |$\binom{6}{2}=15$| possible pairs and all of them are discordant. Moreover, both rankings are strict. Thus, |$n_d=15$| and |$n_c=n_{\pi_T} = n_{\pi_{\widetilde{T}}}=0$|⁠, and we have

$$\tau = \frac{0-15}{\sqrt{(0+15+0)(0+15+0)}} = -1.$$

Results

We are now in the position to study the effects of species extinction, that is, leaf deletions, on rankings obtained from the FP index. We begin by considering circumstances that lead to strict and reversible rankings, that is, circumstances in which the extinction of species completely reverses conservation priorities.

Extinction Scenarios Completely Reversing Conservation Priorities

We start by showing that a strict ranking can only be reversed if the set |$X'$| of deleted leaves contains at least one leaf of each cherry.

 

Theorem 1.

Let |$T$| be a phylogenetic |$X$|-tree with |$|X|\geq 3$|⁠. Let |$ \pi_T$| be a strict and reversible ranking for |$T$| with respect to |$X'\subset X$| and induced subtree |$\widetilde{T}$| on taxon set |$\widetilde{X}=X\setminus X'$|⁠. Let |$c=[x_i,x_j]$| be a cherry of |$T$|⁠. Then, |$X'$| contains at least one of the elements |$x_i$|⁠, |$x_j$|⁠, that is, |$|X'\cap \{x_i,x_j\}|\geq 1$| (note that |$|X'\cap \{x_i,x_j\}|=2$| is possible and that |$X'$| may also contain leaves that are not part of a cherry).

The proof of this theorem is provided in Supplementary Appendix available on Dryad at http://dx.doi.org/10.5061/dryad.jm63xsjbn. As an illustration, consider Figure 1. Here, |$T$| induces a strict and reversible ranking when one leaf per cherry is deleted (in this case, leaves |$x_3$|⁠, |$x_6$|⁠, and |$x_8$| are deleted). If we had kept at least one of those leaves, say |$x_8$|⁠, the resulting ranking would not have been reversible. More explicitly, if we let |$\widehat{T}$| denote the phylogenetic |$\widehat{X}$|-tree with |$\widehat{X}=X \setminus \{x_3,x_6\}$|⁠, we have |$\pi_T(\widehat{X}, FP_T)=(x_1,x_2,x_4,x_5,x_7,x_8,x_9)$| and |$\pi_{\widehat{T}}(\widehat{X},FP_{\widehat{T}}) = (x_5,x_4,x_2,x_1,x_7,x_8,x_9)$|⁠, which shows that the induced rankings are not completely reversed.

So, if we want to find a strict and reversible ranking |$\pi_T$|⁠, then at least one leaf per cherry of |$T$| has to be deleted; otherwise, no suitable edge lengths for |$T$| can exist that induce such a ranking. The following fundamental theorem, however, shows that this necessary condition is even sufficient: For each phylogenetic tree, deleting one leaf per cherry is sufficient for the existence of edge lengths that induce a strict and reversible ranking. Moreover, we can even ensure that the species that has the highest FP index in |$T$| is still present in |$\widetilde{T}$|⁠, where it will have the lowest FP index (as |$\pi_T$| is strict and reversible).

We formalize this in the following main theorem of this section.

 

Theorem 2.

Let |$n\geq 2$| and let |$T=(T_a,T_b)$| be a phylogenetic |$X$|-tree with |$|X|=n$|⁠. Let |$\widetilde{T}$| be the induced subtree on leaf set |$\widetilde{X}\subset X$| that results from |$T$| when we delete precisely one leaf out of each cherry of |$T$| and, if |$n \geq 3$|⁠, suppress the resulting vertices of in-degree 1 and out-degree 1, or, if |$n=2$|⁠, delete the resulting vertex of in-degree 0 and out-degree 1.

Then there exist strictly positive edge lengths |$\lambda_1,\ldots,\lambda_{2n-2}$| for |$T$| such that there is a strict ranking |$\pi_T$| for the leaves of |$T$| concerning the FP index which is reversible with respect to |$\widetilde{T}$| and such that |$\widetilde{T}$| contains the species which has the highest FP index in |$T$| and such that the species with the lowest FP index in |$T$| is not contained in |$\widetilde{T}$|⁠.

In particular, if |$c_T$| denotes the number of cherries of |$T$| and if |$x_1 \in X$| is such that |$x_1=\mathop {\arg \,\max }\limits_{x \in X}{\it FP}_T(x)$|⁠, then we have |${\it FP}_T(x_1)>{\it FP}_T(x_2)>\ldots > {\it FP}_T(x_n)$| and |${\it FP}_{\widetilde{T}}(x_1)<{\it FP}_{\widetilde{T}}(\widetilde{x}_2)< \ldots < {\it FP}_{\widetilde{T}}(\widetilde{x}_{n-c_T})$|⁠, where |$x_1$| as well as |$\widetilde{x}_i$| are contained in |$ \widetilde{X}$| for all |$i=2,\ldots,n-c_T$| and where |${\it FP}_T(\widetilde{x}_i)>{\it FP}_T(\widetilde{x}_j)$| if and only if |$FP_{\widetilde{T}}(\widetilde{x}_i)<{\it FP}_{\widetilde{T}}(\widetilde{x}_j)$|⁠. Moreover, if |$x' \in X$| is such that |$x'=\mathop {\arg \,\min }\limits_{x \in X}FP_T(x)$|⁠, then |$x' \not\in \widetilde{X}$|⁠.

The proof of this theorem (together with additional lemmas required for the proof) is provided in Supplementary Appendix available on Dryad and uses induction on the number of leaves. However, we remark that it is constructive in the following sense: If |$T=(T_a,T_b)$| is a phylogenetic tree with |$n \geq 2$| leaves such that both |$T_a$| and |$T_b$| induce strict and reversible rankings, then the Proof of Theorem 2 establishes a technique to construct a strict and reversible ranking for |$T$| by suitably modifying the edge lengths of |$T_a$| and |$T_b$|⁠. By recursively applying this technique, an edge length assignment yielding a strict and reversible ranking can be found for any given phylogenetic tree |$T$|⁠, regardless of the number of leaves or shape of |$T$|⁠.

Note that while, by Theorem 1, several leaves of |$T$| need to be deleted in order to reverse the entire ranking if |$T$| contains more than one cherry, the following corollary shows that for all values of |$n$|⁠, the extinction of only one species, even the one with the lowest FP index, may be sufficient to cause a strict and reversible ranking (depending on the tree shape).

 

Corollary 1.

Let |$n \in \mathbb{N}_{\geq 2}$|⁠. Then, there exists a phylogenetic |$X$|-tree |$T$| with |$|X|=n$|⁠, namely the caterpillar tree, and edge lengths for |$T$|⁠, such that |$T$| has a strict and reversible ranking with respect to |$\widetilde{X}:=X\setminus \{x\}$|⁠, where |$x=\mathop {\arg \,\min }\limits_{x' \in X}FP_T(x')$|⁠.

 

Proof.

Let |$T$| be the caterpillar tree on |$n$| leaves, that is, |$T$| has precisely one cherry. By Theorem 2, there are edge lengths for |$T$| which assign the smallest |${\it FP}_T$| value to a leaf in a cherry such that, if we delete this leaf, the entire ranking induced by |${\it FP}_T$| gets reversed. □

The Impact of the Extinction of a Single Species

While Corollary 1 implies that for the caterpillar tree the extinction of a single species may completely reverse conservation priorities, for trees that contain more than one cherry the extinction of a single species cannot completely reverse the ordering (due to Theorem 1). In the following, we show, however, that the extinction of a single species, even if it is ranked the lowest, can still cause radical changes in conservation priorities. We begin by showing that given a phylogenetic tree |$T$|⁠, the extinction of the species with the lowest |${\it FP}_T$| value can have the effect that the species with the second-lowest |${\it FP}_T$| value has the highest |${\it FP}_{\widetilde{T}}$| value.

 

Theorem 3.

Let |$T$| be a phylogenetic |$X$|-tree with |$|X|=n \geq 2$|⁠. Then, there exist strictly positive edge lengths |$\lambda_1, \ldots, \lambda_{2n-2}$| for |$T$| such that

• (i)

  the ranking |$\pi_T$| induced by the FP index for the leaves of |$T$| is strict, and

• (ii)

  deleting leaf |$y := \mathop {\arg \,\min }\limits_{x \in X} {\it FP}_T(x)$| from |$T$| results in a strict ranking |$\pi_{\widetilde{T}}$| for tree |$\widetilde{T} := T \setminus \{y\}$| on leaf set |$\widetilde{X} = X \setminus \{y\}$|⁠, for which |$w := \mathop {\arg \,\max }\limits_{x \in \widetilde{X}} {\it FP}_{\widetilde{T}}(x) = \mathop {\arg \,\min }\limits_{x \in \widetilde{X}} {\it FP}_T(x)$|⁠.

In other words, there exist strictly positive edge lengths for |$T$| such that if the species with the lowest |${\it FP}_T$| value goes extinct, the species with the second-lowest |${\it FP}_T$| value has the highest |${\it FP}_{\widetilde{T}}$| value.

The proof of this theorem is provided in Supplementary Appendix available on Dryad, but an example for its implications is depicted in Figure 2. Here, the species with the lowest |${\it FP}_T$| value is taxon |$x_1$|⁠, and the one with the second-lowest |${\it FP}_T$| value is taxon |$x_2$|⁠. However, when |$x_1$| goes extinct, |$x_2$| is the taxon with the highest FP index in the remaining tree. Note that the overall ranking is not completely reversed in this situation in accordance with Theorem 1.

Figure 2.

Tree |$T$| is an example of a tree as described in Theorem 3. Here, |$x_1$| has the lowest |$FP_T$| value and |$x_2$| the second lowest, but |$x_2$| has the highest |$FP_{\widetilde{T}}$| value in the remaining subtree |$\widetilde{T}$| when |$x_1$| is deleted.

As the extinction of the lowest-ranked species can have the effect that the formerly second least important species is ranked highest when the FP indices are recomputed, conservation efforts might need to be reallocated to focus on this species. In the following, we analyze the extinction of any single species, not necessarily (but possibly) the lowest-ranked one, in a little more depth. We first show that if only one species goes extinct and this species is distinct from the highest-ranked species, say |$x^\ast$|⁠, then while there can be changes in conservation priorities, we can at least bound the number of species that will receive an FP index higher than that of |$x^\ast$| and thus require more urgent conservation attention than |$x^\ast$| in the remaining tree (Theorem 4(1)). More precisely, this number is bounded by |$n_a-1$|⁠, where |$n_a$| is the number of leaves in the larger subtree |$T_a$| of a rooted binary tree |$T=(T_a,T_b)$| with |$n \geq 3$| leaves. Note that |$\lfloor \frac{n-1}{2} \rfloor \leq n_a-1 \leq n-2$| (if |$T=(T_a,T_b)$| is such that the number of leaves of |$T$| is as evenly distributed across |$T_a$| and |$T_b$| as possible, we have |$n_a-1 = \lfloor \frac{n-1}{2} \rfloor $|⁠, and if |$T=(T_a,T_b)$| is such that the difference in the number of leaves between |$T_a$| and |$T_b$| is as large as possible (as for example in the case of a caterpillar tree), we have |$n_a-1=n-2$|⁠). This means that the impact of a single species extinction on the conservation priorities of the remaining species directly depends on the shape of the underlying tree and how different its subtree sizes are. However, we then also show that the bound of |$n_a-1$| species receiving a higher FP index than the formerly highest ranked species |$x^\ast$| can be realized in all cases (Theorem 4(2)) (the bound of |$n_a-1$| is true in general, but it is not sharp for all possible edge length assignments; however, there is an edge length assignment that realizes it). Thus, in particular, if |$n_a$| is large, the effect of a single species extinction might require a drastic shift in conservation attention for the remaining species. However, if |$n_a$| is small, then the impact of a single species extinction might be considered less dramatic (even though it could still be the case that almost half of the species require more urgent conservation attention than |$x^\ast$|⁠). Additionally, we remark that a better bound, namely |$n_b-1$|⁠, can be obtained if the deleted taxon is in the smaller subtree |$T_b$|⁠.

 

Theorem 4.

Let |$T=(T_a,T_b)$| be a phylogenetic |$X$|-tree with |$|X|=n\geq 3$| such that |$T_a$| and |$T_b$| have |$n_a$| and |$n_b$| leaves, respectively, where |$n_a\geq n_b$|⁠. Let |$\lambda:E(T)\longrightarrow \mathbb{R}_+$| be some function that assigns all edges of |$T$| positive edge lengths. Let |$x^*:=\mathop {\arg \,\max }\limits_{x \in X} {\it FP}_T(x) $| be the leaf of |$T$| with the highest |${\it FP}_T$| value concerning the edge length assignment of |$\lambda$|⁠. Then, we have:

1. If a leaf |$x'$| other than |$x^*$| is deleted (e.g., the leaf with minimal |${\it FP}_T$| value) to derive a tree |$\widetilde{T}$|⁠, we denote the number of leaves that have a higher |${\it FP}_{\widetilde{T}}$| value than |$x^*$| in |$\widetilde{T}$| by |$m$| and have that |$m \leq n_a-1$| (with |$ \lfloor \frac{n-1}{2}\rfloor \leq n_a-1$|⁠).

2. There exists an edge length assignment |$\widehat{\lambda}$| such that this bound is achieved, that is, |$m= n_a-1$|⁠, and the resulting ranking |$\pi_T$| is strict.

The proof of this theorem can again be found in Supplementary Appendix available on Dryad. For the first part we use the fact that if |$T=(T_a,T_b)$| is a tree with |$n \geq 3$| leaves and some (but not all) leaves from only one of |$T_a$| and |$T_b$|⁠, say |$T_a$|⁠, are deleted, the FP index for all taxa in |$T_b$| remains the same (Lemma 5, Part (i), in Supplementary Appendix available on Dryad), whereas the FP index for all taxa in |$T_a$| strictly increases (Lemma 5, Part (ii), in Supplementary Appendix available on Dryad). The second part of the proof is similar to the proof of Theorem 2. In particular, it provides a constructive way to find an edge length assignment with the claimed properties.

An example to illustrate Theorem 4 is given in Figure 3. Here, the taxon with the highest FP index is taxon |$x_6$|⁠. If we now delete taxon |$x_1$|⁠, which has the lowest FP index in |$T$|⁠, we have that |$n_a-1=3$| leaves receive a higher FP index than |$x_6$| in the resulting tree |$\widetilde{T}$|⁠.

Figure 3.

Tree |$T$| is an example for a tree as described in Theorem 4. Here, |$x^\ast = x_6$| has the highest FP index, but if |$x'=x_1$| (the leaf with the lowest FP index) is deleted, |$n_a-1=3$| leaves (namely, |$x_2$|⁠, |$x_3$|⁠, and |$x_5$|⁠) have a higher FP index than |$x_6$| in the resulting tree |$\widetilde{T}$|⁠.

The Impact of the Edge Lengths

Note that the proofs of the preceding theorems rely on a careful choice of edge lengths. It is thus a natural question to analyze how restrictions on the edge lengths influence the results. For instance, if we assume a molecular clock condition, that is, if we restrict the analysis to ultrametric trees where all leaves have the same distance to the root, what are the worst-case scenarios in this setting? In the special case of caterpillar trees, a molecular clock assumption is beneficial in the sense that the extinction of one or more leaves does not change the ranking order of the remaining leaves.

 

Proposition 1.

Let |$T$| be an ultrametric caterpillar tree on |$X$| with |$|X|=n \geq 3$|⁠, and let |$X' \subset X$| be a subset of the leaves. Let |$\widetilde{T}$| be the induced subtree of |$T$| restricted to the leaves in |$\widetilde{X} := X \setminus X'$|⁠. Then, |${\it FP}_T(x_i) \geq {\it FP}_T(x_j)$| implies |${\it FP}_{\widetilde{T}}(x_i) \geq {\it FP}_{\widetilde{T}}(x_j)$| for all |$x_i,x_j \in \widetilde{X}$|⁠.

We provide a proof of this proposition in Supplementary Appendix available on Dryad. Intuitively, in an ultrametric caterpillar tree, the fewer edges separate a leaf from the root (i.e., the smaller the so-called depth of a leaf), the higher its FP index (in particular, the two leaves in the cherry of a caterpillar tree have the lowest FP index, and the leaf that is adjacent to the root has the highest FP index). Now, if one or more leaves are deleted from a caterpillar tree, the resulting tree is again a caterpillar tree, for which this property still holds. An example is given in Figure 4.

Figure 4.

Ultrametric caterpillar tree |$T$| with |$\pi_T=(x_6,x_5,x_4,x_3,x_2,x_1)$|⁠. If leaves |$x_2$| and |$x_5$| are deleted, the resulting tree |$\widetilde{T}$| is again an ultrametric caterpillar tree and we have |$\pi_{\widetilde{T}} = (x_6,x_4,x_3,x_1)$|⁠. In particular, the remaining leaves appear in the same order in |$\pi_{\widetilde{T}}$| as in |$\pi_T$|⁠, that is, the ranking is not changed by leaf deletions.

So, in the case of ultrametric caterpillar trees, the extinction of species does not influence the ranking order of the remaining leaves. However, as Figure 5 shows, the assumption of a molecular clock does not always imply that the ranking order is unaffected by leaf deletions. This also becomes evident in our simulation study below. A more in-depth analysis of the effects of leaf deletions on the rankings induced by the FP index in the case of ultrametric trees is thus an interesting direction for future research.

Figure 5.

Ultrametric tree |$T$| with |${\it FP}_T(x_1) = {\it FP}_T(x_2)=2.25 < {\it FP}_T(x_3) = {\it FP}_T(x_4) = 2.75 < {\it FP}_T(x_5)=4$|⁠. If we now construct a tree |$\widetilde{T}$| by deleting leaf |$x_2$|⁠, we get |${\it FP}_{\widetilde{T}}(x_3) = {\it FP}_{\widetilde{T}}(x_4) = 2.83 < {\it FP}_{\widetilde{T}}(x_1) = 3.33 < {\it FP}_{\widetilde{T}}(x_5) = 4$|⁠. In particular, |$x_3$| and |$x_4$| are ranked higher than |$x_1$| in |$T$|⁠, while they are ranked lower than |$x_1$| in |$\widetilde{T}$|⁠.

Data Analysis

It is conceivable that the changes in FP rankings due to extinctions that we have studied here are just “theoretical,” and that the problem is not significant with empirical tree data. In order to test the extent of these problems with empirical data, we accessed the free TreeBase database (Piel et al. 2002; Vos et al. 2012) on May 14, 2021 and downloaded all 19,488 trees with up to 100 taxa. We then filtered these trees as follows: We omitted all trees which are unrooted, nonbinary or which do not have branch lengths provided for each edge. We also omitted trees for which all branch lengths are 0. The remaining tree set contained 575 trees. For all these trees, we performed the following two analyses with the computer algebra system Mathematica (Wolfram Research 2017).

1. We calculated the FP values for all taxa, subsequently detected the taxon with the lowest FP value, and deleted it from the list of FP values. The resulting list of FP values was saved as |$originalList_1$|⁠. Then, we deleted the taxon with the lowest FP value and its pending branch also from the original tree. We then recalculated the FP values for the resulting tree. The FP values of this tree were saved as |$newList_1$|⁠. Then, we calculated Kendall’s |$\tau$|⁠: |$\tau(originalList_1,newList_1)$| and saved it in the list |$kendall_1$|⁠. The results of this first study are shown in Figure 6 (black boxplots).

2. We calculated the FP values for all taxa and subsequently detected all cherries. Then, for each cherry, we deleted the taxon with the lowest FP value within the cherry from the list of FP values. The resulting list of FP values was saved as |$originalList_2$|⁠. Afterwards, we also deleted these taxa (the lowest-ranked one of each cherry) and their pending branch from the original tree and recalculated the FP values for the resulting tree. The FP values of this tree were saved as |$newList_2$|⁠. Then, we calculated Kendall’s tau: |$\tau(originalList_2,newList_2)$| and saved it in the list |$kendall_2$|⁠. The results of this study are shown in Figure 6 (gray boxplots).

Figure 6.

The resulting values of Kendall’s |$\tau$| when the leaf with the smallest FP value gets deleted and all FP values get recalculated (black boxplots) and the resulting values of Kendall’s |$\tau$| when of each cherry the leaf with the smallest FP value gets deleted and all FP values get recalculated (gray boxplots). The first boxplots contain all trees with up to 100 taxa from TreeBase, whereas the other boxplots are sorted by the numbers of taxa the respective trees contain.

In both studies, it can be seen that the effects of taxon deletion tend to be less extreme if the tree has more taxa. However, this was to be expected, because, say, a single rank swap between two entries would have a larger effect on Kendall’s |$\tau$| of a tree with few taxa than on Kendall’s |$\tau$| of a tree with many taxa.

Recall that Kendall’s |$\tau$| is 1 precisely if the compared rankings are identical, whereas Kendall’s |$\tau$| is 0 if the compared rankings are uncorrelated. Interestingly, for small trees with up to 20 taxa, some outliers are actually closer to 0 than to 1, that is, their rankings change significantly. One such example is tree |$\textsf{Tr66501}$| from TreeBase, whose Kendall’s |$\tau$| equals 0.333, which is the minimum value observed in the first study.

Note that TreeBase contains trees on a huge variety of species, not all of which are of interest for species conservation programs (and actually, tree |$\textsf{Tr66501}$| on Arabidopsis thaliana is such an example).

However, in order to highlight that there are indeed FP ranking problems concerning trees on species that are of high interest for biodiversity conservation, we explicitly downloaded the tree depicted in Figure 7 from the TimeTree database (Kumar et al. 2017). In this tree of seven species, if one of the two lowest-ranked species, say Cephalorhynchus eutropia, goes extinct, its sister species, which was before also ranked lowest, turns into the second-highest ranked, thereby overtaking three species that were before ranked higher. The relevance of this example becomes evident when you note that C. eutropia, that is, the Chilean dolphin, is listed in Appendix II of the Convention on the Conservation of Migratory Species of Wild Animals (CMS) (CMS Convention on the Conservation of Migratory Species of Wild Animals 2020), which lists all migratory species that have an unfavorable conservation status.

Figure 7.

The species tree of four Cephalorhynchus species and three Delphinus species as downloaded from the TimeTree database (Kumar et al. 2017). Here, |$T$| refers to the tree as depicted, whereas |$\widetilde{T}$| refers to the tree that is obtained when Cephalorhynchus eutropia, one of the two lowest-ranked species, goes extinct. Its sister taxon, Cephalorhynchus commersonii, which also had the lowest rank, then gets the second-highest rank.

So, our studies clearly show that taxon deletion from trees, which happens when species go extinct, can have dramatic effects on the FP index as a ranking criterion. As the comparison between the boxplots in Figure 6 shows, this effect is generally larger the more species go extinct. We included only rooted trees in our studies which were binary and for which all branch lengths were given, but we suspect that similar effects can be seen for nonbinary trees and trees with partial branch lengths as well.

Simulations

Proposition 1 motivated us to analyze the impact of branch lengths in more depth. It is well-known that the so-called Yule or Yule–Harding model (Harding 1971), a pure birth model, leads to ultrametric (“clocklike”) trees. Proposition 1 shows that at least ultrametric caterpillar trees cannot suffer from FP rank swaps when leaves go extinct, which might suggest that having a low death rate (in the Yule model, the death rate is 0) prevents this problem. However, it turns out that actually the opposite is correct, as the following simulations show.

We used the computer algebra system Mathematica (Wolfram Research 2017) to perform two studies. For these studies, we first simulated three sets of trees which were subsequently used for both studies.

For all tree sets, we used |$\lambda=10$| as a birth rate. For the first tree set, we used death rate |$\mu=0$| (which corresponds to the Yule model and produces ultrametric trees), for the second tree set, we used |$\mu=5$| and for the third tree set, we used |$\mu=\lambda=10$|⁠.

Each of the three tree sets contains 300 simulated trees, namely 100 trees each with 10 taxa, 30 taxa, and 50 taxa, respectively.

For all three tree sets, we then performed the same two studies as for the data set presented in the previous section, that is, we first deleted the lowest-ranked leaf from each tree and compared the resulting ranking with the ranking induced by the corresponding subtree of the original tree using Kendall’s |$\tau$|⁠. We then repeated this procedure, but deleted the lowest-ranked leaf from each cherry (instead of only the overall lowest-ranked leaf). The results of the two studies are presented in Figures 8 and 9. Note that both studies show the same overall trends as our analysis of the TreeBase data; namely that the more taxa a tree has, the lower the impact of a few rank swaps; and more leaf deletions tend to cause more rank swaps than a single leaf deletion.

Figure 8.

The resulting values of Kendall’s |$\tau$| when the leaf with the smallest FP value gets deleted and all FP values get recalculated. The first boxplot contains all trees with up to 50 taxa that were simulated (with constant birth rate |$\lambda=10$| and death rates |$\mu=0$|⁠, |$\mu=5,$| and |$\mu=10$|⁠, respectively), whereas the other boxplots are sorted by the numbers of taxa the respective trees contain.

Figure 9.

The resulting values of Kendall’s |$\tau$| when of each cherry the leaf with the smallest FP value gets deleted and all FP values get recalculated. The first boxplot contains all trees with up to 50 taxa that were simulated (with constant birth rate |$\lambda=10$| and death rates |$\mu=0$|⁠, |$\mu=5,$| and |$\mu=10$|⁠, respectively), whereas the other boxplots are sorted by the numbers of taxa the respective trees contain.

However, our simulations show more than that: They show that the higher the death rate of branches in the simulated trees, that is, the more nonultrametric the tree is (and thus the more diverse the branch lengths are), the smaller the damage caused by leaf deletions. We suggest a possible explanation for this observation and discuss it more in-depth in the Discussion section.

Discussion

The FP index is a popular phylogenetic diversity index used to prioritize species for conservation. However, even if a species receives active conservation attention, there is still a risk that it goes extinct. The aim of the present manuscript was thus to analyze the effects of species extinction on the prioritization order obtained from the FP index. More specifically, we analyzed the extent to which the ranking order may change when some species go extinct and the FP index is recomputed for the remaining taxa. On the one hand, we showed that the extinction of one leaf per cherry might completely reverse the ranking. On the other hand, we proved that even the extinction of only the lowest-ranked species in a tree can cause significant changes in the prioritization order. On the positive side, in the case of a single species extinction not involving the highest-ranked species, we also showed that the number of species that require more urgent conservation attention than the formerly most important species in the remaining tree can at least be bounded from above. Here, we saw that the effects of a single species extinction are less dramatic if the underlying tree is “balanced” in the sense that its two maximal pending subtrees are of similar sizes, whereas the impact is more severe if these subtree sizes are very different and the deleted taxon is in the larger subtree. Note that the balance of a tree also played a major role when we showed that the extinction of one leaf per cherry can completely reverse the FP ranking as the number of cherries is in fact also often used to measure the balance of a tree (cf. McKenzie and Steel 2000; Kersting and Fischer 2021). Investigating the impact of the shape of a tree, or more precisely its balance, on the FP index when species go extinct is thus an interesting direction for future research.

Moreover, the present results rely on a particular choice of edge lengths and do not immediately carry over to situations where restrictions on the edge lengths are in place. While we showed in Proposition 1 that the FP ranking on ultrametric caterpillar trees is not affected by species extinction, we also saw that this is not the case for other ultrametric tree shapes. On the contrary, our simulation results obtained subsequently showed that the more ultrametric a tree is, the more sensitive to leaf deletions it tends to be. This can probably be explained by the fact that the FP indices of leaves in trees with a high |$\mu$|-value (i.e., a high death rate) seem to have a higher variation of FP indices than in ultrametric trees: In our simulations, the median variance for birth–death trees with |$\lambda=10$| and |$\mu=0$| was 0.00139859, while the median variance for birth–death trees with the same birth rate |$\lambda$| but death rates |$\mu=5$| and |$\mu=10$| were higher, namely 0.00176227 and 0.00183703, respectively. This higher variation of FP indices amongst the leaves of these trees might imply that not so many of these leaves can simply swap their ranks because of extinctions. If there is less variation in the FP indices, that is, the FP indices are more similar and closer together, their ranks can more easily be swapped when relatively few changes in the tree occur. A second immediate direction for future research would thus be to analyze the effects of species extinction on the prioritization order obtained from the FP index when ultrametric trees are considered more in-depth.

Another interesting direction for future research would be to analyze how other phylogenetic diversity indices, for example, the so-called “Equal-Splits index” (Redding 2003; Redding and Mooers 2006), or prioritization indices based on other aspects of biodiversity such as “feature diversity” or “functional diversity” are affected by species extinctions and whether they are more “robust” than the FP index.

We remark, however, that while we showed that the prioritization order obtained from the FP index might radically change when species go extinct, we do not suggest to disregard the FP index or other phylogenetic diversity indices completely. Our aim was merely to draw attention to these potential “conservation regrets” (i.e., cases where the initial choice of conservation priority might need substantial readjustment after events of species extinction). A way forward in this regard might be to perform a sensitivity analysis prior to conservation decisions to assess the impact of extinction events, for example, of the lowest-ranked species, on the prioritization order, and to adjust conservation attention accordingly. Developing a quantitative measure or test to assess the “robustness” of a phylogenetic diversity ranking for different scenarios (e.g., for the extinction of the lowest-ranked species or the extinction of a highly ranked species despite conservation efforts) is thus another important direction for future research.

Supplementary material

Data and appendix available from the Dryad Digital Repository: http://dx.doi.org/10.5061/dryad.http://dx.doi.org/10.5061/dryad.jm63xsjbn and at: http://mareikefischer.de/SupplementaryMaterial/FP_Index.zip.

Funding

This work was supported by the joint research project DIG-IT! funded by the European Social Fund (ESF), reference: [ESF/14-BM-A55- 0017/19], and the Ministry of Education, Science and Culture of Mecklenburg-Vorpommerania, Germany to M.F.; The Ohio State University’s President’s Postdoctoral Scholars Program to K.W.

Acknowledgments

The authors wish to thank Sophie Kersting for checking the nexus files containing the simulated data for syntax correctness.