I am happy with the changes made to the manuscript regerding my comments.

Yet, I am not convinced by the reply to my comment 2 (about case 2a in the proof of Claim 1). My concern was the possibility of the other two caterpillars being embedded in (exactly one) leaf of N_r that is NOT a leaf of L_Q. Then the constraint “all caterpillars with a leaf embedded into a leaf of L_Q have the same parity as Q” does not apply to these caterpillars. They are not, under my assumption above, embedded in a leaf of L_Q. So unfortunately, I still don't see why this situation is covered in case 2a.

Review of the manuscript: "When three trees go to war", by Leo van Iersel, Mark Jones and Mathias Weller.

In this manuscript, the authors investigate the question of the number of reticulation vertices in a network displaying a given set of $k$ phylogentic trees on $n$ leaves. For the case $k=2$, which was solved in 2005, the answer is $n-2$, meaning that for two phylogenetic trees, there always exists a network with at most $n-2$ reticulation vertices displaying both trees. In this contribution, the authors show that for $k \geq 3$, the answer is at least $(3/2-\epsilon)n$. More specifically, they show that for all $\epsilon>0$, there exists $n>0$ and three trees on $n$ leaves such that any network displaying these three trees has at least $(3/2-\epsilon)n$ reticulation vertices (Theorem 1). To the best of my knowledge, this is the first identified lower bound for this problem. Along the way, they also provide a bound to the number of leaves required in a multi-labelled tree (a kind of leaf-labelled tree in which two or more leaves may get the same label) displaying a given set of three phylogenetic trees (Corollary 1).

The paper is technical in essence, very well written, and I found no major flaw in the reasoning. There are however a few minor technicalities, which the authors may wish to address:

1 - Looking at the induction in the proofs of Lemma 1 and Lemma 2, I think you should state in the Preliminaries that a single isolated network is considered a MUL-network (or a single isolated edge, whichever you prefer). Your current definition implies that a MUL-tree necessarily has two or more leaves, which conflicts with the base case of your induction in both lemmas.

2 - In the proof of Claim 1, case 2: I am wondering about the leaves of $N_r$ that are not in $L_Q$. Should we not have a case 2a', where none of the other two caterpillars are embedded in a leaf of $L_Q$, but are both embedded in a leaf of $N_r - L_Q$? If this situation cannot happen, then the reason is not obvious to me.

3 - At the end of Rule 1, you claim that you never need to remove more token than the quantity present in the token reservoir. However, Claim 2 "only" states that the number of tokens at the end of the process is non negative. Since the proof of Claim 2 actually shows the former (stronger) statement, maybe you could rephrase Claim 2.

4 - At the very end of the proof of Claim 2. You showed that the number of bad leaves is at most $6n^{log_3 2}$, and the token reservoir contains at least $n$ tokens by the time the first withdrawal is made. But each withdrawal removes $2q$ tokens, not only $q$. So unless I am missing something, $n>6qn^{log_3 2}$ is not enough to ensure that we never remove "too many" tokens. Because with $6n^{log_3 2}$ bad leaves, we will remove $12qn^{log_3 2}$ tokens in total, not $6qn^{log_3 2}$.

5 - Some typos:
-First paragraph of section 2: "occurance" -> "occurence" (4 times).
-In Construction 1: "recusively" -> "recursively".
-In the proof of Lemma 1: "let let" -> "let".

Van Iersel, Jones, and Weller construct a family of triples of permutations that give rise to triples of rooted phylogenetic trees which are difficult to embed into a single phylogenetic network. While the problem of embedding two trees has been studied extensively, considering more trees has been regarded much harder. This manuscript yield the first lower bound on the maximum number of reticulations that can be required to embed exactly 3 binary trees. 

The construction of the sequences is not very surprising, but it is hard to estimate the number of reticulations. The authors use multi-labeled trees as an intermediate structure. They first show that many leaves are needed to embed the input trees into a multi-labeled tree, and then they establish that a network with fewer reticulations than claimed would allow a multi-labeled tree with fewer leaves than possible. 

The result is relevant for applications, because there is no biological reason why the embedding of k trees into a phylogenetic network should be restricted to k=2. The proofs are sophisticated and require complicated notation as well as distinguishing many cases. Some time is needed to understand the details of the proofs, but the paper is well written and organised, making it as easy as possible for the reader to follow. In the last section various questions for further research are asked which are all interesting. A natural conjecture would be that the maximum number of reticulations needed for k trees with n taxa would asymptotically be c(k)n where c(k) is the sum of 1/i for i ranging from 1 to k-1. 

In summary, this is an interesting paper that is enjoyable to read and likely to have some impact on future work on the hybridization number problem. 

I found four typos/minor errors that the authors might want to correct, but another round of review is not necessary. 

Figure 2: Y_3 should end with 312645, not 645312. 

l. 146 "let let"

ll. 236 and 243: Use 'monotonically' instead of 'monotonously'.  

l.278: Add 'and' between 'that' and 'leaves'.