For a taxa set S = {S₁, S₂,…,S_n} of size n, a split consisting of two disjoint non-empty subsets of S is denoted by A | B that is, A and B. If A and B contain all the taxa in S, then A | B is called a full split; otherwise, it is called a partial split. In a phylogenetic tree, each edge is a full split that divides the tree into two parts, while in a phylogenetic network, a group of parallel edges with equal length represents a full split. If |A| = 1 or|B| = 1, the split A|B is called a trivial split. For example, the phylogenetic tree in Figure 1A contains five trivial full splits, such as a|bcdef, and three non-trivial full splits de|abcf, bc|adef, and ade|bcf. In general, a split A|B with |A| = m and |B| = n is called an m|n split. In addition, W(A|B) represents the evolutionary distance between taxa groups A and B. If A or B contains more than two taxa, then W(A|B) calculates the distance between the common ancestor of A and B. For example, W(a|de) = 2, W(d|ae) = 1 in Figure 1A , W(a|d) represents the evolutionary distance between taxa a and d, and therefore, through Figure 1A and these definitions, we can get this equation W(a|d) = W(a|de) + W(ae|d).

For an MSA, a simple parsimony-based method is used to estimate the weights of quartets and sextets. For example, if the character in a site is the same for taxa a, b, and c and for taxa d, e, and f, but different for a and d, then the site is defined to support the split abc | def. For any sextet abc | def, its weight W(abc|def) is defined to be the proportion of total number of sites supporting it in the MSA. The weight of a quartet say ab|cd is calculated in a similar way. After all the quartet and sextet weights are obtained, an ever-expanding process is performed based on these weights to all full splits and their weights. As shown in previous literatures (Bandelt and Dress, 1992; Yang et al., 2013), reconstructing a phylogenetic tree or network is equivalent to calculating all the full splits and their weights. Thus, we have obtained the reconstructed tree or network by this process, which could be shown by a software SplitsTree4 (Huson and Bryant, 2006).

As represented by equation W(a|d) = W(a|de) + W(ae|d), there is such an equation W(abc|def) = W(abc|defg) + W(abcg|def), which can be seen as adding a new taxon g to either side of a split abc|def. If W(abc|def) = 0, then W(abc|defg) = 0 and W(abcg|def) = 0. If taxa group A₁ ⊆ A and B₁ ⊆ B, or A₁ ⊆ B and B₁ ⊆ A, we call the split A|B displays A₁|B₁. It is proven in Bandelt and Dress (1992) that W(A|B) ≤ W(A₁|B₁). Therefore, a split with zero weight cannot be further expanded to larger splits with positive weights.

For a taxa set S with size n, there are 10(6n) sextets. We first calculate the weights of all quartets and sextets from the MSA, and then we expand them to get all full split weights using an ever-expanding process. Suppose there is a septet of abc|defg type, we have W(abc|defg) = W(abc|def) − W(abcg|def), and there is a similar equation for W(abcg|def), so the weight of W(abc|defg) can be obtained by similar continuous calculations, as follows.

Combining the above equations, we have

For |B| ≥ 4, taking minimum over all possible cases, we have

When |A|=4 and |B|=4, the weight of the 4|4 split

where A−A′ for two sets A and A′ denotes set difference (subtraction).

For example A={a, b, c, d}, B={e, f, g, h}, there are eight equations for W(abcd|efgh),

For any split A|B with |A| ≥ 4 and |B| ≥ 4, we traverse the elements in A and B and take out four taxa for each calculation. Suppose a, b, c, d ∈ A and e, f, g, h ∈ B, and we have

For any 2|n split of ab|B type with c, d, e ∈ B, we calculate their weight by formula (5) referred in Quartet-Net (Yang et al., 2013),

Finally, for any trivial split of a|S − a type with b, c ∈ S−a in a taxa set S, we calculate the weight as follows (see also Yang et al., 2013):

Formulas (1) – (6) are used to calculate all full splits by decomposing sextet weights iteratively.

QS-Net takes an MSA as input. Suppose that there are n taxa in the taxa set S, which are arranged in the order of 1, 2, 3, …, n. In the initialization step, all triplet, quartet, and sextet weights are calculated directly from the MSAs. We calculate the weights of full splits in the following ways.

By the above procedures, we calculate the weights of all full splits. Similar to Yang et al. (2013), it is usually advisable to filter the non-trivial full splits with very low split weights, which tend to be false positives. In practice, we remove splits with weight less than c% of the average weight, where c is a user-defined threshold setting to be 1 in this study. The output file containing all non-zero full splits and their weights is stored in.NEXU file format, which can be visualized using SplitsTree4 (Huson and Bryant, 2006). The time complexity of QS-Net is O(n¹⁰).

The tree data were generated from Figure 1B with 12 leaves. For brevity, we only listed reconstructed taxa set in the left or right block containing fewer number of taxa ( Supplementary Material: Table S1 ). For example, split bd|acefghijkl was listed as bd. We then normalized each split by the weight of a split successfully constructed by all methods. All trivial full splits were not listed because they can be successfully reconstructed by all five methods. As shown in Table 1 , all five methods can successfully reconstruct all full splits in the 100 runs of the tree data; the accuracy is equal to the experimental bootstrap value divided by the real bootstrap value. The true-positive split result represents all splits in the real phylogenetic history of the simulated data sets. We listed the number of true-positive splits obtained by the five methods on all simulated data sets in Table 2 . If a method can reconstruct the true-positive split once in 100 runs, we determined that the true-positive split can be obtained by this method. In addition to true-positive results, other split results reconstructed by the method are false-positive splits, which typically have very few weight values. Except for Neighbor-Joining, the other four methods reconstructed some false-positive splits (here we only list false-positive splits with a bootstrap value ≥10). For example, Quartet-Net and QS-Net reconstruct two additional split al and ae with bootstrap values of 10 and 26, respectively (see Table 3 ). This is because QS-Net and Quartet-Net methods use the same calculation formula for split of 2|n type. Neighbor-Net identifies 35 false-positive splits with bootstrap value ranging from 10 to 40. These false-positive splits may be caused by some random mutations in the tree data set.

The network data were generated from Figure 2A containing three reticulate events A, B, and C, which can be decomposed into eight feasible underlying trees. A feasible tree can be obtained by cutting off one branch respectively at A, B, and C. For example, we can get an underlying tree by cutting off the three edges qA, mB, and oC in the three reticulate events. The sequence data of a taxon m were generated by concatenating partial sequence data from q and partial sequence data from r. All true splits and splits reconstructed by the five methods are listed in Supplementary Material: Table S2 . The weight of the true split is the sum of the split weights in eight feasible trees. Similarly, we normalized each split with the weight of split ab and multiplied it by 4. As can be seen from the Table 1 , QS-Net and Quartet-Net accurately reconstructed all true splits in all 100 runs, while Neighbor-Net, Split-Decomposition, and Neighbor-Joining failed to reconstruct a large number of true splits. For example, Neighbor-Net failed to reconstruct split gh, fgi, and fgh in more than 90 runs, and Split-Decomposition was unable to reconstruct split bce and bcde in all 100 runs ( Supplementary Material: Table S2 ). Neighbor-Joining obtained even worse result with 16 true splits missing, which is reasonable because Neighbor-Joining only reconstructs trees and retains the strongest compatible splits.

Supplementary Material Table S3 lists all true splits and splits reconstructed from the five methods from the network data. The data set was generated from Figure 2B with a complicated phylogenetic scenario containing five reticulate events. Similarly, the weight of the true split is the sum of the weights of the splits in 32 feasible trees. We normalized each split with the weight of split ce. As can be seen from the Table 1 , only QS-Net method obtains 100% accuracy in all 100 runs, while the other four methods fail to reconstruct some splits in most runs. For example, Quartet-Net failed in reconstructing split fgi and afg in all 100 runs. In addition to the two splits, Neighbor-Net also cannot reconstruct split hj, bcd, and bcde in more than 90 runs ( Supplementary Material: Table S3 ), which happens because Neighbor-Net reduces splits to make the split system planar. Split-Decomposition and Neighbor-Joining still performed poorly. In addition, all methods except for Neighbor-Joining reconstructed some false-positive splits.

The bacterial data set was used in Takahashi and Kryukov (2009) for the analysis of phylogenetic relationships among bacterial species. This data set consists of 36 bacterial genomes containing concatenated sequence of seven genes (16S rRNA, 23S rRNA, gyrB, pyrH, recA, rpoA, and rpoD). The 36 species were divided into three different groups based on different GC content (32–38%, 50–53%, and 64–69%), containing 14, 11, and 11 species, respectively. We took the GC-rich data consisting of 11 bacterial species and a data of 25 species containing both GC-poor and GC-rich bacteria. The MSAs of both data were generated by ClustalW (Larkin et al., 2007) and further fed into to QS-Net, Quartet-Net (Yang et al., 2013), Neighbor-Net (Bryant and Moulton, 2004), Split-Decomposition (Bandelt and Dress, 1992), and Neighbor-Joining (Saitou and Nei, 1987). We ran the program on an MSI laptop with 2.8-GHz processor and 8-GB memory. A comparison of runtime between QS-Net and Quartet-Net on all data sets is shown in Table 4 ; the time statistics for three artificial data sets are the average of all 100 runtimes. The Neighbor-Joining method has the least runtime, and all other three methods can produce results in less than 2 s on all data sets. The reconstructed results were then viewed by SplitsTree4 (Huson and Bryant, 2006). Only three split networks reconstructed by QS-Net and Quartet-Net method on bacterial data set are shown in Figures 3 and 4 .

Figure 3 shows the phylogenetic network of 11 GC-rich bacterial sequence data set by using QS-Net, which is basically consistent with the experimental results in Takahashi and Kryukov (2009). The reconstructed networks of 25 GC-poor or GC-rich (32–38% and 64–69%) sequence data set reconstructed by QS-Net and Quartet-Net are shown in Figures 4A, B , respectively. As can be seen from the figures, the differences between QS-Net and Quartet-Net are quite obvious. There are two distinct parallelograms that represent the reticulate evolution event in the reconstructed network in Figure 4A but not in Figure 4B , which might be neglected by Quartet-Net due to its inability to identify complicated reticulate events. The numbers of full splits reconstructed by the five methods on bacterial data set and the influenza data set are also listed in Table 5 . QS-Net constructs a moderate total number of splits among all comparison methods, probably because the full resolution of taxa is not achieved. In the GC-rich data set, Neighbor-Net constructs three more splits than does QS-Net, while in the GC-poor and GC-rich data set, Neighbor-Net constructs 29 more splits than does QS-Net. In addition, by comparing Figures 3 and 4A , it can be found that GC content may have an important influence on the evolutionary history of bacteria.

The data set consisted of the full genome sequence of 22 H7N9 influenza A viruses aligned by ClustalW (Larkin et al., 2007). These viruses have major relations with the H7N9 virus (Gao et al., 2013) that appeared in China in 2013, which caused human mortality. We estimated the phylogenetic relationships of these 22 influenza A viruses using Quartet-Net and QS-Net. The results are shown in Figures 5A, B , respectively. Table 5 lists the numbers of full splits reconstructed by the five methods on bacterial data set and the influenza data set. General split networks do not actually represent explicit evolutionary events, which makes the interpretation and comparison of reconstruction methods on real data set difficult. So we list the number of splits built by various methods. As can be seen in Table 4 , QS-Net reconstructs 47 full splits, while Quartet-Net reconstructs 45 full splits.

The three viruses that caused human death (A/Shanghai/1/2013, A/Shanghai/2/2013, and A/Anhui/1/2013) were combined. The phylogenetic network indicates that these H7N9 viruses may be derived from the reassortment from influenza subtypes, including avian-origin H7N9 viruses, H9N2 viruses, and H7N3 viruses. In Figure 5B (constructed by QS-Net), the internal region surrounded by H7N9, H7N7, and H7N3 is more complex than Figure 5A (constructed by Quartet-Net), which indicates that the true evolutionary history of H7N9 influenza A viruses is very complex. Of course, the real evolutionary history is unknown, but at least the results constructed by QS-Net are consistent with a few previous findings.