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Effects of discordance between species and gene trees on phylogenetic diversity conservation

Phylogenetic diversity indices such as the Fair Proportion (FP) index are frequently discussed as prioritization criteria in biodiversity conservation. They rank species according to their contribution to overall diversity by taking into account the unique and shared evolutionary history of each species as indicated by its placement in an underlying phylogenetic tree. Traditionally, phylogenetic trees were inferred from single genes and the resulting gene trees were assumed to be a valid estimate for the species tree, i.e., the"true"evolutionary history of the species under consideration. However, nowadays it is common to sequence whole genomes of hundreds or thousands of genes, and it is often the case that conflicting genealogical histories exist in different genes throughout the genome, resulting in discordance between individual gene trees and the species tree. Here, we analyze the effects of gene and species tree discordance on prioritization decisions based on the FP index. In particular, we consider the ranking order of taxa induced by (i) the FP index on a species tree, and (ii) the expected FP index across all gene tree histories associated with the species tree. On one hand, we show that for particular tree shapes, the two rankings always coincide. On the other hand, we show that for all leaf numbers greater than or equal to five, there exist species trees for which the two rankings differ. Finally, we illustrate the variability in the rankings obtained from the FP index across different gene tree and species tree estimates for an empirical multilocus mammal data set.


Introduction
Phylogenetic diversity indices such as the Fair Proportion (FP) index [8,12], Equal Splits (ES) index [13], and Shapley value [7], have become popular tools for prioritizing species for conservation (e.g., Isaac et al. [8], Redding and Mooers [13], Redding et al. [14,15], Vane-Wright et al. [21], Vellend et al. [22]). They quantify 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 prioritization criteria in conservation decisions. While these indices differ in their definitions and properties, they all depend on the underlying phylogenetic tree and its branch lengths. In particular, different phylogenetic trees for the same set of species or identical phylogenetic trees with different branch lengths, may result in different prioritization orders obtained by any of these phylogenetic diversity indices.
Until recently, phylogenetic trees were usually inferred from single genes and the resulting gene trees were assumed to be a valid estimate for the species tree, i.e., the "true" evolutionary history of the species trees under consideration. However, with the advent of rapid sequencing technologies in recent years, it is now common to sequence whole genomes composed of hundreds or thousands of genes, and it is well known that due to various biological phenomena, such as lateral gene transfer and incomplete lineage sorting, conflicting genealogical histories exist in different genes throughout the genome, resulting in discordance between individual gene trees and the species tree (e.g., Maddison [9], Nichols [10], Pamilo and Nei [11]).
In this paper, we analyze the effects of gene and species tree discordance on prioritization decisions based on the FP index. We chose the FP index as it is employed (under the name 'evolutionary distinctiveness') in the so-called "EDGE of Existence" project established by the Zoological Society of London, a conservation initiative focusing specifically at threatened species that represent a high amount of unique evolutionary history ( [8]; see also https://www.edgeofexistence.org/). Moreover, while the FP index itself lacks a direct biological motivation, it coincides with the so-called Shapley value, a result that was first shown by Fuchs and Jin [6]. The Shapley value, originally a concept from cooperative game theory introduced by Lloyd Shapley [16], on the other hand, reflects the average contribution of a species to overall diversity and is characterized by certain desirable axioms (see, e.g., Haake et al. [7]) that motivate its use (and hence that of the FP index) as a prioritization criterion in conservation planning.
Moreover, we focus on gene tree-species tree incongruence caused by incomplete lineage sorting by employing the multispecies coalescent model [2]. Under this model, given a species tree (an ultrametric phylogenetic tree with branch lengths), a distribution on possible gene tree histories evolving within the species tree is obtained, and the species tree as well as the distribution of gene tree histories are the basis for our analysis. More precisely, we compare the prioritization order of taxa induced by the FP index on the species tree with the prioritization order obtained when averaging the FP index over all gene tree histories evolving within the species tree.
The structure of this paper is as follows. We first introduce all relevant notation and concepts. We then analyze circumstances under which the FP index on a given species tree and the expected FP index across all gene tree histories evolving within the species tree induce the same ranking (Theorem 1). Afterwards, we show that for all n ≥ 5, there exists a species tree on n leaves such that the two rankings (i.e., the ranking obtained from the species tree and the ranking obtained from averaging over all gene tree histories) differ (Theorem 2). Finally, we complement our theoretical results by analyzing the variability in the rankings obtained from the FP index across different gene trees and species trees for an empirical multilocus mammal data set. We end by discussing our results and indicating directions for future research.

Notation and preliminaries
To formally state our results, we need some terminology and notation. Throughout this manuscript, X denotes a non-empty finite set (of taxa or species). If not stated otherwise, X = {x 1 , . . . , x n }.
Species trees, gene tree histories, and probabilities under the multispecies coalescent model. A species tree S = (S, τ ) on X = {x 1 , . . . , x n } is a pair consisting of a rooted binary phylogenetic tree S with leaf set X (i.e., a rooted binary tree whose leaves are identified with X) and a vector of interval lengths τ = (τ 0 , τ 1 , . . . , τ n−1 ), where τ i denotes the length of the interval between the i th and (i + 1) th speciation event in S measured in coalescent units. Note that we assume here that no two speciation events in S occur at exactly the same time (this is motivated by the fact that under typical models for species trees, the probability for two speciation events to occur simultaneously is zero). Moreover, note that the length of τ n−1 , i.e., the length of the interval above the root of S, is infinite. Finally, note that the definition of τ as a vector of interval lengths between speciation times implies that S = (S, τ ) is equidistant or ultrametric, meaning that the length of the path from the root of S to any leaf of the species tree is the same. This is also commonly referred to as the molecular clock assumption. Moreover, note that we sometimes refer to the length of the path from the root of S to any leaf as the height of S.
Given a species tree S = (S, τ ), we use the concept of a gene tree history h associated with S to indicate a sequence of coalescent events as well as the populations and time intervals in which they occur. Informally, a gene tree history h can be thought of as a phylogenetic tree with leaf set X evolving within S (Figure 1). Note that the set of gene tree histories associated with S is finite, and we denote it by H(S).
Moreover, we associate with each history h a vector of coalescent times t h = (t 1 , . . . , t n−1 ), where, moving back in time, two distinct lineages coalesce or merge into a single lineage. For example, referring to Figure 1, at time t 1 , the lineages corresponding to x 1 and x 2 merge into a single lineage. Note that there are infinitely many choices for the values t 1 , . . . , t n−1 satisfying the constraints of a history h evolving within S, and in the following we will often integrate over all possible coalescent times consistent with h. Moreover, note that we assume that no two coalescences occur at exactly the same time, i.e., t i = t j for all i, j ∈ {1, . . . , n − 1}. Thus the coalescent times induce a ranking of the interior vertices of a gene tree history h according to their distance from the root, where the interior vertex with the largest distance (i.e., the vertex corresponding to the most recent coalescence event) is assigned rank 1 and the root itself is assigned rank n − 1. In particular, we consider two histories to be distinct if their coalescent times induce different rankings of the interior vertices ( Figure 2). Formally, this corresponds to the notion of ranked gene tree histories [3,4] (similarly, ranked species trees are defined as species tree topologies together with an order on the speciation events). However, for brevity, we speak of gene tree histories or histories for short. In addition, when we refer to a history h, we usually mean h and its associated vector of coalescent times t h as well as the information in which time interval each coalescent event occurred. Note that we explicitly keep track of the time intervals and, next to the induced rankings of interior vertices, distinguish histories based on this information. For instance, considering the top part of Figure 2, the scenario where the coalescent event t 1 occurs in time interval τ 1 (rather than τ 2 ) in the population ancestral to x 1 and x 2 , is considered a different history, h 3 say, that is distinct from h 1 . . , x 5 } and history h evolving within S. Note that S and h depict different evolutionary relationships. Intervals of time between speciation events (horizontal lines) are denoted by the τ i (left side of the figure), while times corresponding to coalescent events (black dots) are denoted by the t i (coalescent times are measured from the most recent speciation event). Circled numbers indicate a ranking on the speciation events with 1 ○ being the most recent and 3 ○ the most ancient speciation event, and S together with this ordering defines a ranked species tree.
For a given species tree S = (S, τ ) and a particular (ranked gene tree) history h ∈ H(S) with associated vector of coalescent times t h , the probability of observing h given S under the multispecies coalescent model can be found by integrating over the joint density of coalescence times, which we denote by f h (h, t h ). We refer the reader to Degnan et al. [3,4] for a careful derivation of this density. Essentially, it is a product of exponential terms arising from the occurrence of exponential distributions as waiting times to coalescent Figure 2: Two distinct histories h 1 and h 2 evolving within the same species tree. Note that h 1 and h 2 have the same topology and their coalescent events occur in the same time intervals of the species tree, but the rankings of the interior vertices of h 1 , respectively h 2 , induced by vectors of coalescent times t h1 and t h2 differ. Moreover, both h 1 and h 2 are such that there are two coalescence events in interval τ 2 . By subdividing this interval into three subintervals, namely [0, t 1 ], [t 1 , t 2 ], and [t 2 , τ 2 ], and multiplying densities contributed by subintervals, we e.g. obtain events. More precisely, for a sample of k lineages from a single population, the waiting time to coalescence is exponentially distributed with rate k 2 . The probability that there is no coalescence for a sample of k lineages in a time interval of length τ is then e −( k 2 )τ . Based on this, the density f h (h, t h ) is obtained by multiplying exponential probabilities across the branches and time intervals of S. If there are m i ≥ 0 coalescences in an interval τ i of S, the interval is subdivided into m i + 1 subintervals and densities contributed by the subintervals are multiplied (again, see Degnan et al. [3,4] for a more rigorous definition; see also Figure 2 for an example). As an example, consider the species tree S = (S, τ ) and the gene tree history h ∈ H(S) depicted in Figure  1. First consider the "left" subtree of S containing leaves x 1 and x 2 . Here, x 1 and x 2 fail to coalesce in an interval of length τ 1 , contributing a factor of e −τ1 to the density f h (h, t h ). These lineages then coalesce at time t 1 within the next interval, contributing a factor of e −t1 to the density. Similarly, for the "right" subtree of S, lineages x 4 and x 5 fail to coalesce in an interval of length τ 2 yielding a factor of e −τ2 . The interval τ 3 then contains a coalescent event at time t 2 , which contributes the term e −3t2 (since three lineages were available to coalesce). The remaining two lineages fail to coalesce for the remainder of this interval, contributing a term of e −(τ3−t2) . Finally, considering τ 4 , the most recent coalescent event occurs at time t 3 , thus contributing a factor of e −3t3 to the density, while the final coalescent event occurs t 4 − t 3 time units later, contributing a term of e −(t4−t3) . Thus, in this case, we have Finally, the limits of integration are determined by the vector of coalescent times t h and the vector of interval lengths τ . In the example discussed above, t 1 can range from 0 to τ 2 , t 2 can range from 0 to τ 3 , t 3 can range from 0 to t 4 , and t 4 can range from 0 to infinity. This leads to Exchangeability. Considering the coalescent process for some species tree S on X and a time interval τ , we call two leaves, x i and x j say, exchangeable (with respect to τ ) if they "enter" interval τ as single lineages.
In particular, from this point onward (back in time), x i and x j are "indistinguishable" and every history h with the property that neither x i nor x j coalesce before τ either contains [x i , x j ] as a cherry, or if such a history h ′ does not contain the cherry [x i , x j ], there exists a history h ′′ which only differs from h ′ by a permutation of x i and x j and such that As an example, for the species tree S shown in Figure 1, leaves x 3 , x 4 , and x 5 are exchangeable with respect to interval τ 3 if they enter this interval as single lineages, as is the case for the history depicted. The history h depicted contains [x 3 , x 4 ] as a cherry, but there also exist histories, h ′ and h ′′ say, where x 3 and x 4 do not form a cherry, but such that h ′ and h ′′ only differ by a permutation of x 3 and x 4 , and such that could be the history identical to h except that the coalescence at time t 2 involves x 3 and x 5 , in which case the "matching" history h ′′ would be the history identical to h except that the coalescence at time t 2 involves x 4 and x 5 ).
This exchangeability property and the idea of pairing matching histories will be crucial for many proofs and we will sometimes also consider exchangeability of subtrees (rather than single leaves) defined in an analogous way.
Lowest common ancestors and subtrees. Given a species tree S (a history h) with root ρ and two leaves x i , x j ∈ X, we refer to the last vertex in S (h) that lies on both the path from ρ to x i as well as the path from ρ to x j as the lowest common ancestor of x i and x j and denote it as lca(x i , x j ). Moreover, given a vertex v of S (h), we refer to the subtree of S (h) rooted at vertex v as a pendant subtree of S (h).
Caterpillar and pseudocaterpillar trees. Two particular tree topologies that will be relevant in the following are the caterpillar tree and the pseudocaterpillar tree ( Figure 3). First, recall that a pair of leaves, say x i and x j , of a phylogenetic X-tree is called a cherry, denoted [x i , x j ], if x i and x j share a common parent. Now, let n ≥ 2. Then, a phylogenetic X-tree on X = {x 1 , . . . , x n } is called a caterpillar tree if and only if it has precisely one cherry. Moreover, if n ≥ 4, a phylogenetic X-tree on X = {x 1 , . . . , x n } is called a pseudocaterpillar tree (adapted from [3] 1 ) if it has precisely two cherries and those two cherries form a pendant subtree of size four. The Fair Proportion index. Let S be a rooted phylogenetic X-tree (not necessarily binary) with root ρ and leaf set X = {x 1 , . . . , x n }, where each edge e is assigned a non-negative length l(e) ∈ R ≥0 . Then, the Fair Proportion (FP) index [8,12] for leaf x i ∈ X is defined as 1 Note that [3] defined pseudocaterpillar trees for n ≥ 5.
where P (S; ρ, x i ) denotes the path in S from the root ρ to leaf x i and n(e) is the number of leaves descended from e. Note that xi∈X F P S (x i ) = e∈S l(e), i.e., the FP index apportions the total sum of branch lengths of S (also referred to as the 'phylogenetic diversity' of X [5]) among the taxa in X. Note that we can calculate the FP index both for a given species tree S as well as for all histories h ∈ H(S) (where in case of the latter, the FP index is expressed in terms of both speciation and coalescent times). We denote these variants as F P S (x i ) and F P h (x i ), respectively. As an example, for the species tree S = (S, τ ) and the history h with its vector t h of coalescent times shown in Figure 1, and the leaf x 3 , we have Note that when the FP index is calculated for a particular history h evolving within a species tree S, it will depend on some or all of the coalescent times t 1 , . . . , t n−1 . As there are infinitely many choices for the values t 1 , . . . , t n−1 that satisfy the constraints of history h evolving within S, in the following we define the expected FP index for leaf x i on history h given the species tree S by integrating over all possible times at which the coalescent events consistent with h may occur. More precisely, we consider where f h (h, t h ) denotes the probability density of history h and t h is the vector of coalescent times for h, and as before, the limits of integration are determined by the particular locations of the coalescent events of h within the species tree. Based on this, we define the expected FP index of leaf x i on species tree S as the expected FP index across all gene tree histories h associated with S. More formally, Rankings. The FP index induces a natural ranking of the elements in X, where taxa are ordered according to their FP index. Here, a ranking r is an assignment of ranking numbers to the elements of X, where for any pair of taxa x i , x j ∈ X, x i either receives a higher or lower ranking number than x j , or the ranking numbers are equal (we then call x i and x j tied). We say that a function f : X → R induces a ranking r f (X) if the ranking number of x i in r f is smaller than the ranking number of for some x i = x j , x i and x j receive the same ranking number. In this case, we use the so-called 'standard competition ranking' (also referred to as '1224' ranking), where tied elements receive the same ranking number, say r, and the subsequent ranking number is r = r + t − 1, where t denotes the number of tied elements with ranking number r. In the following, we will consider the ranking of X induced by the FP index on the species tree, denoted as r (S,FP) (X), and the ranking of X induced by the expected FP index, denoted as r (S,E(FP)) (X), and analyze the relationship between these two rankings.

Results
We compare the rankings induced by the FP index on a given species tree S with the ranking induced by the expected FP index on S. We show that under certain circumstances the rankings always coincide, while there also exist cases for which the rankings differ.

Circumstances under which the FP index on the species tree and the expected FP index induce the same ranking
In this section, we characterize certain tree topologies for which the rankings r (S,FP) (X) and r (S,E(FP)) (X) are guaranteed to coincide. However, we begin by showing that if a species tree S on X = {x 1 , . . . , x n } contains a pendant subtree S ′ on a subset X ′ ⊆ X, X ′ = {x 1 , . . . , x m } with 3 ≤ m ≤ n say, and such that the two maximal pendant subtrees of S ′ consist of m − 1 leaves labeled x 1 , . . . , x m−1 and 1 leaf labeled x m , respectively, then among these m leaves, leaf x m has both the highest FP index on S as well as the highest expected FP index.
Proposition 1. Let n ≥ 3 and let S = (S, τ ) be a species tree on X = {x 1 , . . . , x n } containing a pendant subtree S ′ on 3 ≤ m ≤ n leaves, which in turn consists of a maximal pendant subtree on m − 1 leaves, x 1 , . . . , x m−1 say, and a maximal pendant subtree on one leaf, x m say, ( Figure 4). Then, Proof.
(i) First, note that all edges of S above interval τ * 1 (see Figure 4) contribute the same amount, ∆ F P say, . This completes the first part of the proof.
Note that here (as everywhere in this manuscript) the limits of integration are determined by the particular locations of coalescent events of each history within the species tree. We now analyze the set of histories H(S) more in-depth. In particular, we analyze them with respect to the interval τ * 2 in S. Note that x i enters interval τ * 2 as part of some subtree, T xi say, where the size of T xi can range from 1 to m − 1 leaves, whereas x m enters interval τ * 2 as a subtree of size 1, T xm say.
However, subtrees T xi and T xm are exchangeable in the following sense: Each history h ∈ H(S) either (a) contains T xi and T xm as a "subtree cherry" (i.e., the roots of T xi and T xm have a common parent and the two trees form a pendant subtree (T xi , T xm ) of h); or (b) if a history h ′ does not contain (T xi , T xm ) as a pendant subtree, there exists a history h ′′ identical to h ′ except that the roles of T xi and T xm are exchanged and such that Figure 5). In both cases, let lca(x i , x m ) denote the lowest common ancestor of x i and x m in the corresponding gene tree history, and consider the FP indices of x i and x m .
(a) If h is such that T xi and T xm form a pendant subtree of h with root lca(x i , x m ), all edges of h above lca(x i , x m ) (i.e., all edges in the path from the root of h to lca(x i , x m )) contribute the same amount to F P h (x i ) and F P h (x m ). However, considering the paths from lca( where t lca denotes the time of the coalescence event that leads to lca(x i , x m ), the contribution of edges in the path from lca( where the inequality is strict if T xi contains at least two leaves). In total, this implies that we have (b) Now, if h ′ and h ′′ are two histories that differ only by a permutation of the roles of T xi and T xm , we first note that the lowest common ancestor of x i and x m corresponds to the same vertex lca(x i , x m ) in h ′ and h ′′ (see Figure 5(b)). Moreover, as in case (a), all edges above lca( . Now, as T xi contains at least one leaf, whereas T xm contains precisely one leaf, referring to the notation in Figure 5, we can conclude that the contribution of edges in the path from lca( is greater than or equal to the contribution of the corresponding edges in the path from Moreover, similar to case (a), the contribution of edges in the path from u to x m in h ′′ towards F P h ′′ (x m ) is greater than or equal to the contribution of edges in the path from u to In particular note that the inequalities are strict if T xi contains at least two leaves.
Thus, we can partition the set of histories H(S) into the subset of histories of type (a), in which T xi and T xm form a pendant subtree and we have F P h (x m ) ≥ F P h (x i ), and thus and into the subset of histories of type (b), i.e., into pairs of "matching" histories (h ′ , h ′′ ), where, with the reasoning from above, in particular, recalling that Now, as in both cases the inequalities are strict for all histories h such that T xi contains at least two leaves (and such histories always exist), in summary, we have as claimed. This completes the proof. We now show that the ranking induced by the FP index on a species tree S coincides with the ranking induced by the expected FP index for S if S is a caterpillar or pseudocaterpillar tree. Theorem 1. Let n ≥ 2 and let S = (S, τ ) be a caterpillar species tree on X = {x 1 , . . . , x n }. Then, if In particular, the rankings r (S,FP) (X) and r (S,E(FP)) (X) coincide. The same statements hold if n ≥ 4 and S = (S, τ ) is a pseudocaterpillar species tree.
Proof. First, assume that S is a caterpillar species tree and without loss of generality let S be as depicted in Figure 3. Then, we clearly have Then it follows immediately from Proposition 1 (with x j = x m and In particular, x i and x j form the unique cherry in S. This immediately implies that x i and x j are exchangeable and thus and all inequality and equality signs are "preserved" when considering the expected FP index, we clearly have r (S,FP) (X) = r (S,E(FP)) (X). This completes the first part of the proof. Now, assume that S is a pseudocaterpillar species tree with n ≥ 4 leaves as depicted in Figure 3. In this case, we have In both cases, x i and x j form a cherry and are clearly exchangeable. We thus have as claimed. It thus only remains to consider the case that x j ∈ {x 3 , x 4 } and x i ∈ {x 1 , x 2 }. Without loss of generality, assume x j = x 3 and we now note that H(S) can be partitioned into five disjoint subsets: (i) The subset of histories, H 1 (S) say, for which both the coalescence involving x 1 and the coalescence involving x 3 occur in interval τ 3 or later. In this case, x 1 and x 3 are exchangeable and we can conclude that (ii) The subset of histories, H 2 (S) say, for which both the coalescence involving x 1 and the coalescence involving x 3 occur in interval τ 2 . By the symmetry of the 4-leaf subtree of S containing leaves (iii) The subset of histories, say H 3 (S), for which the coalescence involving x 1 occurs in interval τ 1 and the coalescence involving x 3 occurs in interval τ 2 or later. In this case, x 1 enters the interval τ 3 as part of a subtree, T x1 say, containing precisely two leaves, whereas x 3 enters the interval τ 3 as part of a subtree, T x3 say, containing at most two leaves. As in the proof of Proposition 1, T x1 and T x3 are exchangeable in the sense that each history h ∈ H 3 (S) either (a) contains a pendant subtree (T x1 , T x3 ); or (b) if a history h ′ ∈ H 3 (S) does not contain (T x1 , T x3 ) as a pendant subtree, there exists a history h ′′ identical to h ′ except that the roles of T x1 and T x3 are exchanged and such that t h ′ = t h ′′ and . Moreover, as x 1 coalesces strictly before x 3 , for each history in H 3 (S) the pendant edge incident to x 3 is strictly larger than the pendant edge incident to x 1 . Using this and the exchangeability of T x1 and T x3 together with the fact that T x3 contains at most two leaves, whereas T x1 contains precisely two leaves, we can thus conclude that The last two remaining subsets are the subsets, H 4 (S) and H 5 (S) say, where H 4 (S) is the subset of histories for which the coalescence involving x 1 occurs in τ 2 and the coalescence involving x 3 occurs in τ 3 or later, and H 5 (S) is the subset of histories for which the coalescence involving x 1 occurs in τ 3 or later and the coalescence involving x 3 occurs in τ 2 . By the symmetry of the 4-leaf subtree of S  Taking all cases together, this implies that E[F P (x 3 ) − F P (x 1 )|S] > 0 and thus E[F P (x 3 )|S] > E[F P (x 1 )|S] as claimed. Now, as F P S (x n ) > F P S (x n−1 ) > . . . > F P S (x 4 ) = F P S (x 3 ) > F P S (x 2 ) = F P S (x 1 ) and all inequality and equality signs are preserved when considering the expected FP index, we clearly have r (S,FP) (X) = r (S,E(FP)) (X). This completes the proof.
As all species trees with at most four leaves are either caterpillar or pseudocaterpillar trees, Theorem 1 immediately leads to the following corollary: τ ) is a species tree on X = {x 1 , . . . , x n } with n ≤ 4, the rankings r (S,FP) (X) and r (S,E(FP)) (X) coincide.
3.2 On the existence of species trees with different rankings induced by the FP index and the expected FP index for all n ≥ 5 After having analyzed some circumstances under which the FP index calculated for a given species tree and the expected FP index across all gene tree histories associated with this species tree induce the same ranking, in this section we show that for all leaf numbers greater or equal to 5, there exist species trees for which the two rankings differ.
Theorem 2. For all n ∈ N ≥5 , there exists a species tree S = (S, τ ) on X = {x 1 , . . . , x n } such that S contains two leaves, x i and x j say, with F P S ( In particular, S is such that the rankings r (S,FP) (X) and r (S,E(FP)) (X) are different.
The strategy to prove this theorem is to show that there exists a species tree S = (S, τ ) on precisely 5 leaves for which the rankings r (S,FP) (X) and r (S,E(FP)) (X) differ. We then construct a species tree S ′ on n > 5 leaves containing S as a subtree and show that the rank swap in S is preserved in S ′ . For this purpose, we will decompose all histories evolving within S, respectively S ′ , by "cutting" them at a certain height determined by the species tree. We now explain this procedure in more detail.
Let S = (S, τ ) be a species tree of height η and let h ∈ H(S) be a history evolving within S. Let 0 < η ′ ≤ η. By cutting history h at height η ′ , we mean subdividing all edges of h present at time η ′ into two new edges, thereby disconnecting h. In case there is an interior vertex v of h present at time η ′ , we also subdivide it into two copies, one of which forming the root of a subtree of h with the outgoing edges, and the other being a new leaf for the incoming edge. This procedure will subdivide h into a forest of several trees, one of which, say h top , will contain the root of h. We call h top the top part of h concerning cut height η ′ and we refer to the other trees, say h 1 , . . . , h k , in the forest as the chopped parts of h concerning cut height η ′ . Moreover, if we attach to each leaf of h top the number of descending leaves it had in h by edges of length 0 (note that the resulting tree need not be binary and may contain degree-2 vertices), we derive the extended top part of h concerning cut height η ′ , which we denote by h top ext . Whenever there is no ambiguity concerning the cut height, we drop the explicit mention of η ′ . As an example, consider the species tree S and the history h evolving within S depicted in Figure 1 and let η ′ = τ 0 + τ 1 + τ 2 . Then cutting h at height η ′ leads to the decomposition of h into its (extended) top and chopped parts as depicted in Figure 6. We now show that given a cut height 0 < η ′ ≤ η, the FP index of a taxon x i on history h can be calculated by considering the decomposition of h into its extended top part and chopped parts concerning cut height η ′ . Lemma 1. Let S = (S, τ ) be a species tree on taxon set X = {x 1 , . . . , x n } and let η = τ 0 + . . . + τ n−2 denote its height. Let h ∈ H(S) be a history evolving within S. Moreover, let 0 < η ′ ≤ η, and let h top ext be the extended top part of h concerning cut height η ′ and let h 1 , . . . , h k be the chopped parts of h. Then, for each x i in X, we have where h i is the chopped part of h containing x i (note that the root of h i might have out-degree 1) and v xi is one of the newly attached leaves of h top ext whose parent is an ancestor of x i in h.
This completes the proof.
Now, considering the expected FP index across all gene tree histories associated with a species tree S and given a cut height 0 < η ′ ≤ η, as a consequence of Lemma 1 and using linearity of the expectation, we obtain the following decomposition of Equation (3): In the following, we will refer to the first summand in Equation (5) With this we are now finally in the position to prove Theorem 2. Proof of Theorem 2. We first show that the statement holds for n = 5. Let S = (S, τ ) be as depicted in Figure 1 and set τ = (τ 0 , τ 1 , τ 2 , τ 3 ) = (2.114, 0.1269, 0.00293, 0.37684). Then, it can easily be seen that F P S (x 3 ) ≈ 2.36944 > 2.367335 = F P S (x 1 ). Moreover, we used the computer algebra system Mathematica [23] to explicitly enumerate all gene tree histories in H(S) and calculate the expected FP indices of all taxa according to Equation (3) (the full expressions in terms of (τ 0 , τ 1 , τ 2 , τ 3 , τ 4 ) are given in the Appendix). In particular, This completes the proof for n = 5.
In the following, we will construct a species tree S ′ on n > 5 leaves containing S as a pendant subtree such that the rank swap of taxa x 1 and x 3 is preserved in S ′ . However, before we describe this construction in more detail, we remark that if we cut all histories in H(S) at the height of S, i.e., at height η = τ 0 +τ 1 +τ 2 +τ 3 , using Mathematica [23] we obtained the following expected chopped FP indices for taxa x 1 and x 3 and the choice of τ given above: In particular, the order of x 1 and x 3 induced by the expected chopped FP index coincides with the order induced by the overall expected FP index and contradicts the order induced by the FP index on the species tree. Moreover, considering the extended top parts of all histories h ∈ H(S) with respect to cut height η, using Mathematica [23] we calculated the distribution of the number of new leaves that are attached to the last ancestor of x 1 , respectively x 3 , in the top part h top of each history h (again for the choice of τ given above). Let n 1 be the number of leaves attached to the last ancestor of x 1 and let n 3 denote the number of leaves attached to the last ancestor of x 3 . Then, n 1 ∈ {1, 2} (if a history h is such that x 1 and x 2 coalesced before time η, n 1 will be two; otherwise n 1 will be one) and n 3 ∈ {1, 2, 3} (where n 3 = 3 if h is such that all of x 3 , x 4 , x 5 coalesced before time η, n 3 = 2 if x 3 coalesced with x 4 or x 5 but all three taxa did not coalesce, and n 3 = 1 if x 3 did not coalesce) and we have the following distribution (Table 1): Table 1: Distribution of n 1 and n 3 as described in the proof of Theorem 2. Keeping these findings in mind, we now construct a species tree S ′ = (S ′ , τ ′ ) on n > 5 leaves as follows. We take any tree topology S ′ on n − 4 leaves and replace one of the leaves by S. For τ ′ , we keep the interval lengths as induced by τ and choose all remaining interval lengths such that S ′ = (S ′ , τ ′ ) is an ultrametric species tree. We now argue that S ′ has the properties that First, as S replaced a leaf of S ′ , all edges on the path from the root of S ′ to the root of S contribute the same amount, say c, to the FP index of any leaf in {x 1 , . . . , x 5 }. Moreover, no edge lengths within the subtree S of S ′ were changed. Hence, we have F P S ′ (x i ) = F P S (x i ) + c for all x i ∈ {x 1 , . . . , x 5 } and thus in particular In order to show the second assertion, we now consider the decomposition of the expected FP index into the expected chopped FP index and expected top FP index as given in Equation (5) with respect to S ′ and with respect to cut height η = τ 0 + τ 1 + τ 2 + τ 3 (i.e., with respect to the same cut height as discussed for S). If we cut all histories h ∈ H(S ′ ) at height η, the expected chopped FP indices for taxa x 1 , . . . , x 5 on S ′ are identical to the ones observed when cutting all histories evolving within S at height η. In particular, we have It remains to consider Instead of calculating these expected values explicitly, we will consider where v xi is one of the newly attached leaves of h top ext whose parent is an ancestor of x i in h. We thus need to analyze h∈H( For this purpose, we partition the set H(S ′ ) of histories associated with S ′ into two disjoint subsets: (a) the set of histories whose induced top parts contain the cherry [l 1 , l 3 ], and (b) the set of histories whose induced top parts do not contain the cherry [l 1 , l 3 ]. In either case, this will lead to six subcases for the extended top part histories depending on the number of leaves attached to l 1 and to l 3 , respectively.
(a) We first consider the subset of histories in H(S ′ ) whose induced top parts contain the cherry [l 1 , l 3 ].
Consider such a top part and denote the time of coalescence of l 1 and l 3 by t 13 . Now, note that each such fixed top part gives rise to a distribution of extended top parts obtained by attaching n 1 ∈ {1, 2} leaves to l 1 and n 3 ∈ {1, 2, 3} leaves to l 3 (see Figure 7), where n 1 and n 3 are distributed according to Table 1 (due to the fact that we cut all histories in H(S ′ ) at height η, i.e., at the same cut height for which this distribution was obtained when cutting all histories in H(S)). Moreover, note that n 1 and n 3 are clearly independent. Referring to Figure 7, among the histories in H(S ′ ) whose induced top parts contain the cherry [l 1 , l 3 ] and the time of coalescence of l 1 and l 3 is t 13 , a proportion of P (n 1 = 1) · P (n 3 = 1) of extended top parts is of type (i), a proportion of P (n 1 = 1) · P (n 3 = 2) is of type (ii) and so forth. Now, considering these types of extended top parts more in-depth, we observe the following: • If the extended top part of a history h is of type (i) or (v), we have F P h top < 0. However, as type (ii) has probability P (n 1 = 1) · P (n 3 = 2) ≈ 0.218, whereas type (iv) has probability P (n 1 = 2) · P (n 3 = 1) ≈ 0.201, the proportion of histories with extended top parts contributing +t13 2 is larger than the proportion of histories with extended top parts contributing −t13 2 . In summary, we can thus conclude that  Table 1. Using the same enumeration as in case (a), we thus have the following subcases: (i) n 1 = n 3 = 1; (ii) n 1 = 1 and n 3 = 2; (iii) n 1 = 1 and n 3 = 3; (iv) n 1 = 2 and n 3 = 1; (v) n 1 = 2 and n 3 = 2; (vi) n 1 = 2 and n 3 = 3.
Referring to this enumeration scheme and Figure 8 and fixing the time of coalescence leading to lca(x 1 , x 3 ), among the matching pairs of histories in H(S) with induced top parts h top and h top , respectively, a proportion of P (n 1 = 1) · P (n 3 = 1) of pairs of extended top parts is of type (i), a proportion of P (n 1 = 1) · P (n 3 = 2) is of type (ii) and so forth. Similar to case (a), we now analyze these types more in-depth, and observe the following: • For pairs of matching histories ( h, h) with extended top parts of type (i) or (v), we have • For pairs of matching histories ( h, h) with extended top parts of type (iii) or (vi), we have • For pairs of matching histories ( h, h) with extended top parts of type (ii), we have whereas for pairs of matching histories ( h, h) with extended top parts of type (iv), we have However, as in Case (a), pairs of type (ii) have a higher probability than pairs of type (iv), and thus the proportion of matching pairs of histories with extended top parts contributing a positive value is larger than the proportion of histories with extended top parts contributing the same negative value.
As we can partition the subset of histories in H(S ′ ) whose induced top parts do not contain the cherry [l 1 , l 3 ] in pairs of matching histories ( h, h) as described above, we can thus also conclude that Taking Equations (6) and (7) together, we obtain as claimed. This completes the proof.
Remarks. We conclude this section with a few remarks. First, note that the proof of Theorem 2 establishes a slightly stronger result than the statement of Theorem 2. More precisely, we not only prove the existence of one species tree S on n ≥ 5 leaves with the desired property, but in fact establish this result for a family of such trees for n large enough, as we did not fix the tree on n − 4 leaves used in the construction, such that any tree on n − 4 leaves can be used.
Second, note that in the proof of Theorem 2, we used a very particular choice of τ leading to F P S (x 3 ) > F P S (x 1 ) and E[F P (x 3 )|S] < E[F P (x 1 )|S] for the species tree S on n = 5 leaves depicted in Figure 1. However, there are infinitely many numerical values for τ with this property and we depict slices of the corresponding regions of parameter (branch length) space in Figure 9. Note, however, that these regions were solely plotted by considering F P S (x 3 ) > F P S (x 1 ) and E[F P (x 3 )|S] < E[F P (x 1 )|S] for the species tree S on n = 5 leaves depicted in Figure 1   Third, we note that there exist two additional species trees on 5 leaves, S ′ and S ′′ say, with the same topology as the species tree S used in our construction, but with a different order of speciation events (see Figure 10). Expressions for the FP indices on the species trees S ′ and S ′′ and the respective expected FP indices are given in the Appendix. It is an interesting question whether a rank swap can also occur for these two species trees. In case of the species tree S ′ , we can answer this affirmatively. For instance, for τ = (1.0000, 0.0050, 0.0075, 0.0100), we have F P S ′ (x 3 ) ≈ 1.01583 > 1.01375 = F P S ′ (x 1 ), but For the species tree S ′′ , on the other hand, we conjecture that a rank swap is not possible, but leave a proof or counterexample to this conjecture to future research.
Finally, we remark that there is a curious connection between some of the results presented in this manuscript and results obtained by Degnan et al. [4,3] in relation to the existence of so-called anomalous ranked gene trees (ARGTs), whereby a ranked gene tree history that does not match the ranked species tree can have a greater probability than the matching ranked gene tree. More precisely, Degnan et al. [4] showed that ARGTs do not exist for 3-taxon or 4-taxon species trees, and Degnan et al. [3] showed that more generally ARGTs do not exist for caterpillar and pseudocaterpillar species trees. This directly parallels Corollary 1 and Theorem 1 in the present manuscript, which state that the ranking induced by the FP index on the species tree and the ranking induced by the expected FP index cannot differ in these cases. Moreover, Figure 10: Species trees S ′ and S ′′ that have the same unranked topology as the species tree S depicted in Figure 1, but a different ranked topology. In particular, they depict a different order of speciation events (compare the circled numbers to see this).
the species tree S depicted in Figure 1, i.e., the species tree we used in the proof of Theorem 2 to show that there are species trees for which the rankings differ, is exactly the ranked species tree used by [4,3] to prove the existence of ARGTs for 5-taxon trees. As indicated above, rank swaps for the FP index exist for at least one other ranking of this species tree topology (namely for the ranked species tree S ′ shown in Figure 10) and it would be interesting to see if this again parallels or maybe contrasts the case of ARGTs. Based on the characterization of species trees producing ARGTs given by [3] it is unfortunately not possible to decide if the two species trees S ′ and S ′′ induce ARGTs.

Empirical data
In order to illustrate the variability of prioritization orders obtained from the FP index for empirical data, we considered the multilocus mammal data set of Song et al. [17]. As the original collection of gene trees for the data was criticized in the literature for methodological errors, we used those from a re-analysis of Springer and Gatesy [18]. These data consist of 447 gene trees for 37 species (33 of which are mammals and four of which are outgroup species; a list of species names and their abbreviations can be found in Table 2 in the appendix). 3 As the gene trees provided by Springer and Gatesy [18] were unrooted, we rooted them using Gallus gallus as the outgroup. We then calculated the FP index for all 37 species on all 447 gene trees. In addition, we calculated the average FP index of each species across the 447 gene trees. We also calculated the FP index for all species using two methods: the species tree estimation method SVDQuartets [1] as implemented in the PAUP* package [20] and the maximum likelihood method for the concatenated data using RAxML version 8.2.10 [19]. Maximum likelihood branch lengths were computed for both trees under the GTR+Γ model in PAUP*, additionally enforcing a molecular clock. For reproducibility, we provide the exact commands used in the appendix. Finally, the FP indices on the 447 gene trees and the two species trees as well as the average FP indices were transformed into rankings and a boxplot summarizing the results was generated ( Figure 11).
As evident from the boxplot, there is a huge variability of ranking positions across the 447 gene trees for most species. In particular, there are species such as Sus scrofa (Sus), whose ranking positions span the entire range of ranking positions possible or are close to it (as in the case of Callithrix jacchus (Cal), Equus caballus (Equ), or Macaca mulatta (New)). On the other hand, there are some species such as Gorilla gorilla (Gor), Homo sapiens (Hom), Pan troglodytes (Pan), and Pongo pygmaeus (Pon) that show comparatively little variability in their ranking positions and consistently receive low ranks. These four species also show the interesting property that the ranks obtained from the average FP index across the 447 gene trees and the ranks obtained from the two species tree estimates all coincide except for one case (for Pon, the rank from the mean over gene trees differs slightly from the rank on the two species tree estimates). In general, it is interesting to note that the average FP index and the FP index on the two species tree estimates yield similar ranks in many cases. In particular, the rank of the SVDQuartets tree almost always coincides with either the rank based on the average FP index or the rank based on the concatenated data. However, there are nevertheless some interesting rank swaps. For instance, considering species Procavia capensis (Pro) and Tupaia belangeri (Tup), based on the average FP index, Tupaia belangeri receives are higher rank than Procavia capensis, but on both species tree estimates, Procavia capensis receives a higher rank than Tupaia belangeri. This last observation highlights that our theoretical results on possible rank swaps between the FP index on a species tree and the expected FP index across all gene trees associated with it (Theorem 2) might occur in practice and need to be considered in the process of conservation planning.

Concluding remarks
The FP index is a popular phylogenetic diversity index used to prioritize species for conservation. It assesses the importance of species for overall biodiversity based on their placement in an underlying phylogenetic

Rank
Rank from mean over gene trees Rank from SVDQuartets species tree Rank from RAxML species tree (concatenated) Figure 11: Boxplot of the ranking positions obtained from the FP index for the multilocus mammal data set of Song et al. [17] consisting of 447 gene trees for 37 species (a table listing the full scientific and common names of all species can be found in the appendix). In addition, the ranking position obtained from the average FP index across the 447 gene trees (diamonds), the ranking position obtained from the FP index for a species tree estimated with SVDQuartets [1] (stars), and a species tree estimated with RAxML version 8.2.10 [19] for the concatenated data (circles) are depicted.
tree. As such, the prioritization order obtained from the FP index heavily depends on the topology and branch lengths of the phylogenetic tree used. In particular, different phylogenetic trees for the same set of species might lead to different prioritization orders.
In this note, we analyzed the variability of prioritization orders obtained from the FP index for different phylogenetic trees on the same set of species in the context of gene tree species tree discordance. More precisely, we compared the ranking obtained from the FP index on a given species tree with the ranking obtained from the expected FP index across all gene tree histories evolving within the species tree under the multispecies coalescent model. On one hand, we showed that the two prioritization orders are identical for all trees with at most 4 leaves, and more generally for all caterpillar and pseudocaterpillar trees. On the other hand, we showed that for all n ≥ 5, there exists a species tree on n leaves such that the two rankings are different. We ended by illustrating the variability in ranking orders obtained from the FP index for an empirical multilocus mammal data set. Here, we observed a high variability in ranking positions for most species, indicating that if conservation decisions are based on single genes, the prioritization order may be very different depending on which gene is used. We also observed rank swaps between the average FP index across gene trees and the FP index on two species tree estimates, empirically underpinning our theoretical results stated in Theorem 2.
Our work suggests several broad questions that might be interesting to explore in the future. From a theoretical point of view, it would be interesting to analyze if there are classes of tree topologies other than the caterpillar and pseudocaterpillar trees for which the FP index on the species tree and the expected FP index across all gene tree histories necessarily induce the same prioritization order. Moreover, in cases for which there are differences between the two prioritization orders, it would be interesting to assess how different the two rankings can be in the most extreme cases. In particular, we remark that our construction for the existence of a species tree on n ≥ 5 leaves leading to two different ranking orders relies on a switch in ranking positions among the leaves of a certain 5-taxon subtree, and if n is large and this is the only difference between the two rankings, this might be considered a minor difference. It would thus be interesting to see if there are other constructions that lead to more severe differences between the two rankings. From an empirical point of view, it would also be interesting to study the variability of the prioritization orders obtained from the FP index for different gene trees and species tree estimates for a broader class of data sets, in particular for data sets containing conservation-relevant species, i.e., species at high risk of extinction. More generally, as both our theoretical results as well the analysis of the empirical data set show that different prioritization orders may be obtained from considering the FP index on a given species tree, on a particular gene tree, or even on all possible gene trees associated with the species tree, an immediate question that arises is the question of which evolutionary information should be used as the basis for conservation decisions. Should these decisions be based on the species tree, on a collection of gene trees, or taking into account a combination of both? We leave all of these questions to future research, but remark that particularly answering the last question will likely require joint efforts from theoreticians and empiricists.
In the following, we give expression for the FP indices on the species tree and the expected FP indices for the species trees S, S ′ , and S ′′ on 5 leaves considered in the main part of this manuscript (Figures 1 and  10). The expected FP indices were obtained using Mathematica [23] to enumerate all gene tree histories in H(S), respectively H(S ′ ) and H(S ′′ ).
• Expected FP indices: B Species name abbreviations used in Figure 11 C Details on species tree estimation for Section 4 In order to obtain species tree estimates for the multilocus mammal data set of [17], we downloaded the full sequences of the 447 genes from https://datadryad.org/stash/dataset/doi:10.5061%2Fdryad.3629v (file gene447_final.nexus).