Are the Duplication Cost and Robinson-Foulds Distance Equivalent?

Abstract

In the tree reconciliation approach for species tree inference, a tree that has the minimum reconciliation score for given gene trees is taken as an estimate of the species tree. The scoring models used in existing tree reconciliation methods include the duplication, mutation, and deep coalescence costs. Since existing inference methods all are heuristic, their performances are often evaluated by using the Robinson-Foulds (RF) distance between the true species trees and the estimates output on simulated multi-locus datasets. To better understand these methods, we study the relationships between the duplication cost and the RF distance. We prove that the gap between the duplication cost and the RF distance is unbounded, but the symmetric duplication cost is logarithmically equivalent to the RF distance. The relationships between other reconciliation costs and the RF distance are also investigated.

1. Introduction

With more and more genomes being fully sequenced, species tree inferences from multilocus datasets have become one of the basic bioinformatics tasks in comparative and evolutionary biology (Dunn et al., 2008). Reconstructing a species tree from a multi locus dataset requires reconciling the phylogenetic information contained in the genes. The challenge for such an analysis is that the genes sampled from the same set of species often contain conflicting signals and thus have discordant gene trees. Although some of these conflicting signals can be due to systematic errors occurring in phylogenetic analysis, they often reflect gene-specific mutational events. During evolution, recombination, gene duplication, gene loss, incomplete lineage sorting, or horizontal gene transfer, all might occur in a gene family (Fitch, 1970; Goodman et al., 1979; Maddison, 1997; Pamilo and Nei, 1988).

Existing methods for inferring species trees from gene trees can be divided into two broad categories: parametric methods (Ané et al., 2007; Kubatko et al., 2009; Liu and Pearl, 2007) and tree reconciliation methods (Warnow, 2013). Parametric methods adopt gene duplication or incomplete lineage-sorting models and infer species trees using either likelihood or Bayesian framework. These approaches have strong statistical foundation, but they can be computationally expensive. Therefore, tree reconciliation methods are often used in phylogenetic studies on a genomic scale (Burleigh et al., 2011; Katz et al., 2012; Sanderson and McMahon, 2007).

Gene tree reconciliation methods take a collection of gene trees and output the tree that has the smallest reconciliation score for the gene trees as an estimate of the species tree. The scoring models used in these methods include the duplication cost (that is, the number of duplications), the mutation cost (that is, the number of gene duplication and loss events), the deep coalescence cost (that is, the number of incomplete lineage sorting events), and the Robinson-Foulds (RF) distance (Bansal et al., 2010; Bayzid et al., 2012; Chang et al., 2013; Chaudhary et al., 2013; Than and Nakhleh, 2009; Wehe et al., 2008; Yu et al., 2011). Since finding the parsimony tree from gene trees is NP-hard for the reconciliation scoring models (Bryant, 1997; Ma et al., 2000; Zhang, 2011), existing methods all compute a species tree by navigating through the part of the species tree space guided by the reconciliation score. The performances of these heuristic methods are usually evaluated using the RF distances between true species trees and their estimate output on simulation datasets (Chaudhary et al., 2013; Than and Nakhleh, 2009; Yang and Warnow, 2011). Hence, uncovering the relationships between the RF distance and the reconciliation costs is important to design good gene tree reconciliation methods. Unfortunately, many of these relationships are incompletely known. In particular, the relationship between the RF distance and the duplication cost is rather elusive, as both are bounded above by a linear function in the number of taxa. In this work, we mainly investigate this relationship. We consider all the reconciliation costs as different metrics in species tree spaces.

The rest of this article is divided into three sections. The basic concepts, notions, and the various reconciliation costs are introduced in Section 2. Several results are presented in Section 3. We first show that the gap between the duplication cost and the RF distance is unbounded. On the other hand, the symmetric duplication cost and the RF distance are proved to be logarithmically equivalent. A similar relationship between the gene loss cost and the RF distance is obtained. We also examine the distribution of the sizes of terraces for these metrics. We conclude with suggestions for future work in Section 4.

2. Concepts and Notions

2.1. Species trees

A species tree T over a set of species X is a rooted tree in which one node is designated as the root, all edges are oriented away from the root, and the species in X are one-to-one mapped to leaves. We use V(T) and E(T) to denote the sets of nodes and (directed) edges in T, respectively. For , v is the parent of u and, equivalently, u is a child of v if . A node is called a leaf if it does not have any child. In this article, we focus on binary species trees in which every nonleaf node has two children. We also use the following notation:

• • p(u) denotes the parent of a nonroot node u in a tree;

• • Ch(u) denotes the set of two children of a nonleaf u in a tree;

• • V_lf(T) denotes the set of leaves in T;

• • V_it(T) denotes the set of internal (i.e., nonleaf) nodes in T;

• • T(u) denotes the subtree rooted at , which consists of u and its descendants of u;

• • P_T(−, u) denotes the unique path from the root to u in T;

• • T|_U denotes the subtrees induced by a subset , whose sets of nodes and branches are and , respectively.

For , v is an ancestor of u and, equivalently, u is a descendant of v, denoted by , if v is in P_T(−, u). We write if the relationship is uncertain, but we know that v = u or . Therefore, it is not hard to see that v is a leaf (terminal node) if and only if for any . For , lca(U) denotes the latest common ancestor (lca) of the nodes in U.

2.2. The RF distance

Let X be a set of species, and let denote the set of all species tree over X. Consider . A species is called the label of the corresponding leaf , written l(u). Conversely, we define l⁻¹(x) = u. For , we define:

called the (species) cluster of u. Define

Consider two species trees S′ and S″ over X. They are identical if and only . Therefore, the dissimilarity of S′ and S″ can be measured by the number of clusters that are found in one but not in the other, that is,

called the RF distance (Semple and Steel, 2003). Since both S′ and S″ have |X| − 1 internal nodes and their roots' cluster is X, The following inequalities hold:

2.3. Tree reconciliation costs

Consider . Since each leaf in these trees has a unique label in X, there is a one-to-one map between V_lf(S′) and V_lf(S″). Such a correspondence can be extended into a map from V(S′) to V(S″), λ_S′,S″, as:

The duplication cost (DPC), C_dup(S′, S″), is defined as |Dup(S′, S″)|, where

The nodes in Dup(S′, S″) are called duplication nodes.

Proposition 1. For .

Proof. Since S′ ≠ S″, there exists having the following properties (Fig. 1):

FIG. 1.

Illustration of the proof of Proposition 1. The cluster of is not found in S′, but the clusters of the descendants of u are all in S′. If such that , then lca(t₁, t₂) is a duplication node under λ_S′,S″.

• (i) C_S″(u) ≠ C_S′(v) for any ;

• (ii) for each descendant t of u, that is, , C_S″(t) = C_S′(v) for some .

Let Ch(u) = {u₁, u₂}. By the above assumption, C_S″(u_i) = C_S′(t_i) for some for i = 1, 2. Since C_S″(u) is not a cluster in S′, t₁ and t₂ must not be siblings in S′, that is, p(t₁) ≠ p(t₂). Let lca(t₁, t₂) = v and Ch(v) = {v₁, v₂} such that and . Since t₁ and t₂ are not siblings, either t₁ ≠ v₁ or t₂ ≠ v₂. If t₂ ≠ v₂, λ_S′,S″(v₂) is in P_S″(−, u). If t₁ ≠ v₁, λ_S′,S″(v₁) is in P_S″(−, u). Thus, λ_S′,S″(v₁), λ_S′,S″(v₂), or both are in P_S″(−, u), the path from r(T″) to u.

If . If . Therefore, v is a duplication node and hence C_dup(S′,S″) ≥ 1.

Let have the largest distance from the root of S′. Then, x must have two leaf children, x₁ and x₂. By definition, λ_S′,S″(x₁) ≠ λ_S′,S″(x) ≠ λ_S′,S″(t₂). Hence, x is not a duplication node when S′ is mapped onto S″. Since S′ has |X| − 1 internal nodes, S′ has at most |X| − 1 − 1 duplication nodes. Therefore, C_dup(S′,S″) ≤ |X| − 2.   ■

For , we use λ_S′,S″(e′) to denote the path from λ_S′,S″(u) to λ_S′,S″(v). We further use |λ_S′,S″(u, v)| to denote the number of the non-end nodes in λ_S′,S″(e′). |λ_S′,S″(u, v)| = 0 if λ_S′,S″(u) = λ_S′,S″(v). The gene loss cost, C_loss(S′, S″), is defined as:

where χ_u = 1 if u is a duplication node such that |λ_S′,S″(u, û)| ≠ 0 for some and χ_u = 0 otherwise.

If S′ is a gene tree and S″ is a species tree, C_dup(S′, S″) and C_loss(S′, S″) are the numbers of gene duplications and losses occurring in the duplication history of the genes represented by the leaves in S′ defined by the lca map (Goodman et al., 1979).

For , define:

Clearly, is in DC(e″) if and only if e″ is an edge in λ_S′,S″(e). The deep coalescence cost, C_dc(S′,S″), is defined as:

The deep coalescence cost C_dc(S′, S″) is introduced to measure the dissimilarity of S′ and S″ that are caused by incomplete lineage sorting for a gene tree S′ and a species tree S″ (Maddison, 1997).

Finally, we note that all the reconciliation costs introduced above are not a symmetric metric in general. Hence, the symmetric duplication cost (SDPC) of S′ and S″, defined as C_dup(S′, S″) + C_dup(S″, S′), is used to study species tree inference (Ma et al., 2000).

3. Results

3.1. Unbounded gap between the DPC and RF distance

For two different trees over the same set of species, both the duplication cost and half of the RF distance are in the same range from 1 to n − 2, where n is the number of species involved in the trees. Therefore, the following question naturally arises:

Are the DPC and RF distance metrically equivalent?

and, equivalently, does there exist c₁ ≥ c₂ > 0 such that

for any ?

Consider such that C_rf(S₁, S₂) = 2(|X| − 2), that is, . By definition, the parent of two sibling leaves in S₁ is not a duplication node under λ_{S1, S2}. Since such an internal node always exists in S₁, we have:

However, surprisingly, there are many pairs of trees S₁ and S₂ in satisfying that C_rf(S₁, S₂) = 2(|X| − 2) but C_dup(S₁, S₂) = 1, where |X| ≥ 3. For example, we consider a class of complete binary species trees, as illustrated in Figure 2. Let n = 2^k for some k ≥ 1 and . We set T to be the rooted complete binary tree with n unlabeled leaves. A_n denotes the species tree over X obtained from T by labeling the i-th leaf (counting from left to right) with i (Fig. 2). B_n denotes the species tree obtained from T by labeling the i-th leaf with φ(i) that is defined as:

FIG. 2.

Complete binary species trees A_n and B_n for n = 2³. C_dup(A_n, B_n) = 1 but C_rf(A_n, B_n) = 2(n − 2).

for .

Proposition 2. (1). C_rf(A_n, B_n) = 2(n − 2).

• (2). C_dup(A_n, B_n) = 1.

• (3). The DPC and the RF distance are not equivalent.

Proof.

(1). For any nonroot , either , or . In B_n, the (2i − 1)th and (2i)th leaves are siblings, labeled with i and n/2 + i, respectively, where i ≤ n/2. Therefore, for ,

Combining these two facts together, we conclude that only the roots of A_n and B_n have the same cluster X. Hence, C_rf(A_n, B_n) = 2(n − 2), as both trees have n − 1 internal nodes.

(2). Consider . Notice that for some i and k. For the left and right children, u₁ and u₂, of u,

If u is in the left subtree of A_n, then and

If u is in the right subtree of A_n,

Hence, the root r of A_n and its two children are mapped to the root of B_n, implying that r is a duplication node.

If u is in the left subtree of A_n such that , then,

This implies that u is not a duplication node. Similarly, all the nodes in the right subtree are not duplication nodes. Therefore, C_dup(A_n, B_n) = 1.

(3). This statement follows directly from (1) and (2).   ■

Although C_dup(A_n, B_n) = 1, we have that C_dup(B_n, A_n) = n/2 − 2. However, for any fixed large k > 0, there exist G and S such that when X is large enough. For example, we take an integer m and define

Consider the two trees G and S over X defined in Figure 3. It is easy to see that C_rf(G, S) = 2(m² − 2). The lca map λ_G,S maps different nodes in G_i to different nodes in S for each i. Furthermore, it maps the roots of and their ancestors to the root of S. Thus, C_dup(G, S) = m − 1. Similarly, we can also show that C_dup(S, G) = m − 1. Hence, for any k < m, we have .

FIG. 3.

Trees G and S over m² species satisfying that C_rf(G, S) = 2(m² − 2), but C_dup(S, G) = C_dup(G, S) = m − 1.

This example shows that even the symmetric duplication cost (SDPC) is not equivalent to the RF distance in the species tree space. However, they are weakly equivalent.

3.2. Logarithmic equivalence of the SDPC and RF distance

Consider . For the sake of simplicity, we use the following notations:

• • μ denotes the lca map from S′ to S″, that is, μ = λ_S′,S″;

• • ψ = λ_S″,S′;

• • Dup_μ(S′, S″) denotes the set of duplication nodes in S′ under μ.

Lemma 1.

For such that C_rf(S′, S″) = 2(|X| − 2), and .

Proof.

We prove the first fact by mathematical induction. By symmetry, the second fact holds.

Consider . It has two children, u₁ and u₂. There are three possible cases.

Case 1.

u₁ and u₂ are leaves.

Since S′ and S″ do not have any common node cluster except X, the leaves, w₁ and w₂, that have the same labels as u₁ and u₂ are not siblings in S′. By definition, μ(w_i) = u_i and ψ(u_i) = w_i for i = 1, 2. Let v = lca(w₁, w₂). By assumption, ψ(u) = lca(ψ(u₁), ψ(u₂)) = lca(w₁, w₂) = v. Assume that the children of v are v₁ and v₂ such that . Since w₁ and w₂ are not siblings, we have that v₁ ≠ w₁, v₂ ≠ w₂, or both. Without loss of generality, we may assume that v₁ ≠ w₁. Since μ(w₁) = u₁, which is a leaf, and μ(v₁) ≠ u₁, μ(v₁) is either equal to u or an ancestor of u, that is, μ(v₁) is in the path P_S″(−, u). If v₂ = w₂, μ(v) = lca{μ(v₁), μ(v₂)} = μ(v₁) (Fig. 4A), as μ(v₂) ( = u₂) is a child of u and hence a descendant of μ(v₁). If v₂ ≠ w₂, μ(v₂) is in P_S″(−, u) and hence μ(v₁) and μ(v₂) are comparable under , as they are in the path from the root to u (Fig. 4B). If , μ(v) = μ(v₂). If , μ(v) = μ(v₁). Therefore, .

FIG. 4.

Illustration of the proof of Lemma 1 in Case 1. Here, u has two leaf children, u₁ and u₂, ψ(u) = v, and ψ(u_i) = w_i for i = 1, 2. (A) Here, w₂ is a child of lca(w₁, w₂), but w₁ is not. In this case, μ(v) = μ(v₁). (B) Both w₁ and w₂ are not the children of v. The children, v₁ and v₂, are mapped into the path from the root to u under μ.

Case 2.

Only one of u₁ and u₂ is a leaf.

This case is similar to Case 1. Let and . Assume w₁ is the leaf corresponding to u₁ in S′, that is, and assume that . If w₁ is a descendant of ψ(u₂), then . If w₁ is not a descendant of ψ(u₂), that is, , we let v = lca(w₁, ψ(u₂)). Then, v = ψ(u) and μ(v) is in P_S″(−, u). Since S′ and S″ do not have any common cluster except X, μ(ψ(u₂)) is also in P_S″(−, u). If w₁ is a child of v, then μ(v) = lca(w₁, μ(ψ(u₂))) = μ(ψ(u₂)). If w₁ is not a child of v, we assume that v₁ is the child of v such that . Then μ(v₁) is in the path P_S″(−, u) and is comparable with μ(ψ(u₂)). We obtain that μ(v) = μ(v₁) if and μ(v) = μ(ψ(u₂)) otherwise. Therefore, ψ(u) is v, a duplication node under μ.

Case 3.

Both u₁ and u₂ are not leaves.

Assume for i = 1, 2. If ψ(u) = ψ(u_i) for some i, it is obvious that .

If ψ(u) ≠ t₁ and ψ(u) ≠ t₂, we have that ψ(u) = lca(t₁, t₂). Let lca(t₁, t₂) = v and Ch(v) = {v₁, v₂} (Fig. 1). We prove that v is a duplication node under μ by considering the following two cases.

Case 3.1.

t_i ≠ v_i for i = 1, or 2. Assume t₁ ≠ v₁. Since t₁ and u₁ have different node clusters, μ(t₁) is an ancestor of u₁ and hence μ(v₁) is in the path P_S″(−, u). Since μ(v) = lca(μ(v₁), μ(v₂)), μ(v) must be in P_S″(−, u) and thus is equal to μ(v₁) if . It is equal to μ(v₂) if , as μ(v₂) is also in P_S″(−, u) in this case.

Case 3.2.

t₁ = v₁ and t₂ = v₂. Since C_rf(S′, S″) = 2(|X| − 2) and the roots of S′ and S″ have the same species cluster, but C_S″(u₁) ≠ C_S′(v₁). Hence, μ(v₁) is in the path P_S″(−, u). Similarly, μ(v₂) is also in P_S″(−, u). This implies that they are comparable under . Hence, μ(v) = lca(μ(v₁), μ(v₂)), equal to μ(v₁) if and equal to μ(v₂) otherwise. Hence, .   ■

For , we use μ⁻¹(v) to denote the set of nodes that are mapped to v under μ. If v is the root of is a subtree rooted at the root of S′. If v is a nonroot internal node in is a union of disjoint subtrees of S′.

Lemma 2.

Let be a nonroot node and m duplications nodes be mapped into P_S″(−, p(v)). Then, μ⁻¹(v) is a union of m + 1 subtrees of S′ at most.

Proof.

Let v be a nonroot internal node in S″. Assume that μ⁻¹(v) is a union of . Let T_i be rooted at . Then, x_i's are incomparable, that is, and for i ≠ j. This implies that is a tree with k leaves in which k − 1 internal nodes are binary and other internal nodes have only one child.

Let y be a binary internal node that has two children in . Clearly, y = lca(x′, x″) for some . Assume that y₁ and y₂ are the children of y in S′ such that and . Since μ(x′) = μ(x″) = v, both μ(y₁) and μ(y₂) are in the path P_S″(−, p(v)). This implies that μ(y₁) and μ(y₂) are comparable under . Therefore, we have either or . If the former is true, μ(y) = μ(y₁). If the latter is true, μ(y) = μ(y₂). Thus, y is a duplication node under μ.

Since all the k − 1 binary internal nodes in are duplication nodes under μ, k − 1 ≤ m. Therefore, k ≤ m + 1.   ■

Lemma 3.

Let . If there exist nonroot internal nodes and such that C_S′(u) = C_S″(v). Let denote the tree obtained from S′ by replacing S′(u) with a single leaf and the tree obtained from S″ by replacing S″(v) with a single leaf with the same label. Then,

Theorem 1.

For .

Proof.

By Lemma 3, we may just assume that only the root clusters of these two trees are identical, that is, C_rf(S′, S″) = 2(|X| − 2). Let μ and ψ be defined as above. Assume that

Since each v_i is a duplication node with respect to the lca map ψ from S″ to S′,

We further assume that μ⁻¹(v_i) contains q_i nodes and d_i out of them are duplication nodes, for each 1 ≤ i ≤ k. Note that is a union of subtrees of S′ and every internal node in such a subtree is a duplication node with respect to μ.

Assuming that is a union of t_i subtrees of . Let T_j have b_j internal nodes, each of which is a duplication node. Since T_j is binary, it has exactly b_j + 1 leaves, each of which is not a duplication node. Therefore, we have:

and

Since there are at most C_dup(S′, S″) − d_i duplications nodes that are mapped in the path P_S″(′, p(v_i)), by Lemma 2, we have that t_i ≤ C_dup(S′, S″) − d_i + 1 if v_i is a nonroot node in S″, and it is trivial for the root of S″ for which t₁ = 1. This implies that

Hence, by Inequalities (8) and (9),

By Proposition 1,

and thus

Using Inequality (10), we obtain the following fact:

   ■

Theorem 2.

For arbitrary species trees S′, S″ over the same set of species,

Therefore, the SDPC and the RF distance are logarithmically equivalent

Proof.

By Theorem 1,

and thus

On the other hand, by Inequality (7) and Lemma 3,

This implies the following inequality:

   ■

3.3. The relationships between the RF distance and the gene loss cost

Consider . Let such that . Assume and Ch(v) = {v₁, v₂}. We have that is not empty for i = 1, 2. Since C(u) ≠ C(v), C(u) is a proper subset of C(v). This implies that λ_S′,S″(v) is in P(−, u). We consider the following two cases.

Case 1. Only one of v₁ and v₂ is mapped into P(−, u). We assume that v₁ is mapped to a descendant of u, that is, . Then, (p(u), u) is an edge in λ_S′,S″(v, v₁). Since λ_S′,S″(v₂) is in P(−, u), v₂ has a descendant such that (p(u), u) is an edge in . Thus, there is one extra lineage in (p(u), u) at least.

Case 2. Both v₁ and v₂ are mapped into P(−.u). Similar to Case 1, v_i has a descendant such that (p(u), u) is an edge in for i = 1, 2. Thus, there is one extra lineage in (p(u), u) at least. This proves:

Conversely, if |X| ≥ 3 and C_rf(S′,S″) = 2(|X| − 2), by theorem 6 in Than and Rosenberg (2013),

It is easy to see that the result in Lemma 3 is also true for the deep coalescence cost. Combining all these facts together, we have:

This implies the logarithmic equivalence of the RF distance and the deep coalescence cost.

For any , C_loss(S′, S″) = C_dc(S′, S″) + 2C_dup(S′, S″), proved in Zhang (2011). By Inequality (11),

On the other hand, by Inequalities (7) and (11) and Lemma 3,

Therefore, the RF distance and the gene loss cost are also logarithmically equivalent.

3.4. The size distribution of terraces in species tree space

Since species tree space grows exponentially with the number of taxa, understanding the landscape of the tree space is important in designing good heuristic algorithms that navigate through parts of this space guided by the parsimony score of a tree to infer species trees. Given a metric defined in the species tree space, the set of species trees with the same metric score from a species tree is called a terrace (Sanderson et al., 2011).

Here, we examine the size distribution of the DPC and RF terraces. Formally, a terrace defined by a species tree S for the DPC score c is . Similarly, we can define the concept for the RF distance. We adopt the Furnas rank (Furnas, 1984) to display the size distributions of these terraces. It is a complete order in the species tree space defined as follows. For if one of the following three conditions holds:

• 1. The left subtree of S′ has fewer leaves than the left subtree of S′.

• 2. , but .

• 3.  and are identical, but , where and are the right subtrees of S and S′, respectively.

Figure 5 displays the size distributions of terraces with respect to the DPC (left panel) and RF distance (right panel) for the species tree space over 10 species. We observe the following facts:

FIG. 5.

The size distribution of terraces defined by the DPC and the RF distance. All 98 topologies of the species tree space over 10 species are listed in increasing Furnas rank along the X-axis. Each color bar indicates the portion of species trees in a terrace that have the same score (given inside a circle in the corresponding band) with the corresponding species tree.

• • The two terraces defined by the two largest values contain over 95% of the trees for the RF distance, independent of the Furnas rank of the reference tree.

• • In contrast, the terraces defined by the middle values of the DPC are large, whereas the terraces defined by the extreme DPC cost are small. The size of the terrace of the largest DPC value increases slightly as the Furnas rank of the reference tree increases.

Figure 6 displays the size distributions of terraces with respect to the SDPC and gene loss costs. Interestingly, these size distributions are quite similar to each other. They indicate that SDPC and gene loss costs are more sensitive to the topology of the trees than the DPC and the RF distance.

FIG. 6.

The size distribution of the terraces defined by the symmetric duplication cost (left panel) and the gene loss cost (right panel). Ninety-eight topologies of the species trees over 10 species are listed in increasing Furnas rank along the X-axis. Each color bar indicates the portion of species trees in a terrace.

4. Conclusion

All the tree reconciliation methods have high accuracy in inferring true species trees from real gene trees data. However, recently, Chaudhary et al. (2013) demonstrated by simulation that the RF-based parsimony approach often outperforms the reconciliation methods that use the gene duplication or mutation costs. A recent study of Yang and Warnow (2011) also suggests that the duplication-cost-based approach has poorer performance at large. One of our results is that the DPC and the RF distances are not equivalent. This provides a plausible explanation for the facts just mentioned. The weak equivalence of the RF distance and the gene loss cost also gives a clue to why the mutation cost is desirable for inferring species trees from a collection of gene trees (Chaudhary et al., 2013; Yang and Warnow, 2011).

Mathematical properties of the tree reconciliation costs have been studied in recent years. The deep coalescence cost has the Pareto property (Lin et al., 2012), meaning that a node cluster that appears in every input gene tree must also appear in the parsimony species tree. Than and Rosenberg (2013) report the formulas for the number of trees that achieve the maximum deep coalescence cost for a fixed gene or species tree. Górecki and Eulenstein (2014) presented a linear time algorithm for identifying a tree that maximizes the deep coalescence cost for a fixed gene or species tree. Zhang (2011) studied the relationship among the duplication cost, gene loss cost, and deep coalescence cost. The relationships between the nearest neighbor interchange distance and the reconciliation costs are also investigated (Bansal et al., 2009; Luo et al., 2011; Than and Rosenberg, 2013). However, many of mathematical properties of the reconciliation costs are incompletely known. For example, the distributions of these costs remain unclear for the uniform and Yule-Harding distribution (Harding, 1971) of species trees. In contrast, the distribution of the RF distance is much more clear. It is asymptotically Poisson with different means for several underlying species tree distributions (Steel and Penny, 1993). Our preliminary studies on terraces for the DPC and RF distances indicate that the distribution of the DPC seems to be different from that of the RF distance. Therefore, the following problems are open for future research:

• 1. Is the distribution of the DPC (respectively, SDPC) asymptotically normal?

• 2. Is the distribution of the gene loss cost asymptotically normal?

Solutions to these problems are definitely informative for studying species tree inference.

Acknowledgments

L.X.Z would like to thank Jian Chen for helpful discussions. The work was supported by the Singapore MOE (Tier-1 FRC grant R146-000-177-112).

Author Disclosure Statement

No competing financial interests exist.