**Last updated:** 2018-01-05

**Code version:** 582bc0b

Polyploid species played a major role in the evolution of plants. In this paper, we focus on present all possible phylogenetic networks from a multilabeled tree that are guaranteed to have minimal number of interaction nodes.

*Figure 2 from original paper*

Based on Fig.2 in this paper, we can see (b)(c)(d) are all phylogenetic networks that exhibit (a), however, we will use efficient algorithm to draw a phylogenetic network like (d) rather than (b) and (c).

From Fig.3 in this paper, we can see subtrees \(T_{(u)}\),\(T_{(v)}\) and \(T_{(w)}\) are maximal inextendible. Fours useful concepts are subtree, equivalent, inextendible and maximal inextendible. In fact, we will focus on how to find maximal inextendible subtrees from a given MUL tree in actual algorithm. Note that the definition of **inextendible** is not clear for me, so I borrowed another one in the paper of Huber KT and Moulton to get a better understanding this terminology.

**inextendible**: suppose \(T\) is MUL tree, for every vertex \(v \in V(T)\) that is not the root of T we denote the parent of \(v\) by \(\bar v\), suppose \(T'\) is a sub MUL tree with vertex \(v\), we say \(T'\) is **inextendible** if there existed another sub MUL tree \(T''\) with root vertex \(w\) so that \(T''\) is isomorphic to \(T'\), and \(T(\bar v)\) is not isomorphic to \(T(\bar w)\).

*Figure 3 from original paper*

So there would be a ** contradiction** in the statement of inextendible definition. Take a look at Fig.3 again, whether on earth each subtree having leaves labeled with

`b`

and `c`

is inextendible or not ?The core of this algorithm showed in Fig.4 is to merge and prune maximal inextendible subtrees and its equivalent subtrees, this process is repeated until a network is obtained that contains no repeated labeled leaves.Unfortunately, I could understand how do they find the maximal inextendible subtrees by height list \(H\) and code \(c(v)\) in the initial step.

A noticeable limitation is when MUL tree contains polytomies showed in Fig.6,using this presented constructing methods would lead to several different phylogenetic networks.

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