 at the EA Play event ahead of E3 2017.
Kent McFann
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Kent McFann, is the head of the Entertainment and Theatre Technology program, chair of the Department of Theatre Arts, and a theatre professor at Saddleback College. He has been involved in theatre and the entertainment industry in varying capacities since he was a child when he spoke the line, “Walk on water?” in the musical The King and I. When he decided acting wasn’t his future, he pursued the many avenues possible in technical theatre.
Mr. McFann received his Bachelor of Fine Arts degree in Set Design from the California Institute of the Arts and his Master of Fine Arts degree in Set Design from Purchase College (aka SUNY Purchase).
He has designed scenery for numerous theatre shows, the most notable of which was for an off-off Broadway original production, Cross Your Heart directed by A. Dean Irby. Mr. McFann has also worked professionally as a television Art Director. His work in television includes work on music award shows, game shows, music specials, and reality shows. He was nominated for a 2000 Primetime Emmy Award for Outstanding Art Direction for HBO’s “Cher Live from the MGM”. Mr. McFann’s other design work includes storefront interior design and museum design. He is also the co-inventor of the MUT PinSplitter and MUT Handle.
Although Mr. McFann is proud of all his work, one of his favorite roles is that of volunteer designer and technical consultant for the free afterschool drama program at Carl Hankey K-8 in Mission Viejo.\section{Introduction}

Leaf-labelled trees, also called phylogenetic trees, are used to represent inferred evolutionary histories of sets of species \cite{sempleSteelBook}. Due to different data sets  and methods that are used to infer  such histories, even phylogenetic trees that have been reconstructed for the same set of species often differ. To quantify the dissimilarities between trees, several tree metrics are commonly used that compute distances between two phylogenetic trees (for a recent review, see~\cite{st2017shape}). Moreover, to summarise the information that two or more phylogenetic trees have in common, maximum agreement subtrees (MASTs) have become a popular tool. Historically, the concept of a MAST was introduced by Finden and Gordon \cite{finden1985} as a way of measuring  similarities among an arbitrary number of phylogenetic trees. Intuitively, an agreement subtree for a collection ${\mathcal P}$ of phylogenetic trees is a leaf-labelled tree ${\mathcal M}$ that can be embedded in each tree in ${\mathcal P}$. If ${\mathcal M}$ has the maximum number of leaves over all agreement subtrees for ${\mathcal P}$, then ${\mathcal M}$ is a maximum agreement subtree  (formal definitions are given below).

Various aspects of MASTs have been well studied over the years. While the problem of computing a MAST for at least three phylogenetic trees is NP-hard in general~\cite{amir1997maximum}, several polynomial-time  algorithms have been developed to compute a MAST for two binary phylogenetic trees \cite{cole2000n,goddard1994agreement,steel1993kaikoura}. 
In terms of a lower bound on the size of a MAST, Martin and Thatte \cite{martin2013maximum} have shown that two unrooted binary phylogenetic trees on $n$ leaves have a MAST on $\Omega(\sqrt{\log n})$ leaves. Their result improves on a lower bound of $\Omega(\log\log n)$ leaves that was  previously established by Steel and Sz\'ekely \cite{steel2009improved}.
Markin \cite{markin2018extremal} has recently closed the gap between lower and upper bound asymptotics by showing that the minimum MAST for two unrooted binary phylogenetic trees has  $\Theta(\sqrt{\log n})$ leaves.  Turning to phylogenetic trees of a particular shape, Martin and Thatte
\cite{martin2013maximum} have  investigated the size of a MAST for two balanced rooted binary phylogenetic trees on $n=2^m$ leaves. In particular, they have shown that a MAST for two such trees has at least $2^{\beta m}$ leaves, where $\beta \sim 0.149$ and, subsequently, conjectured that  two balanced rooted binary phylogenetic trees on $n$ leaves have a MAST with at least $n^{\frac 1 2}$ leaves. The main result of this paper, disproves their conjecture for infinitely many values of $n$ via a family of counterexamples. 

For completeness, it is also worth mentioning that 
lower and upper bounds on the expected size of a MAST for two phylogenetic trees that are generated under the uniform or Yule-Harding distribution have been established in \cite{bernstein2015bounds,bryant2003size,misra2019bounds}. Interestingly, in the context of this paper, Misra and Sullivant~\cite{misra2019bounds} have shown that the expected size of a MAST for two balanced rooted binary phylogenetic trees on $n$ leaves is $\Theta(n^{\frac 1 2})$ under the uniform distribution. 


To formally state the main results, we require some terminology. A {\it rooted phylogenetic $X$-tree} ${\mathcal T}$ is a rooted tree with leaf set $X$ and with no degree-two vertices, except for the root which has degree at least two. For technical reasons, if $|X|=1$, we additionally allow ${\mathcal T}$ to consist of the single vertex in $X$, in which case, this vertex is the root as well as the leaf of ${\mathcal T}$. The {\it size} of ${\mathcal T}$ is $|X|$. If $|X|=1$ or ${\mathcal T}$ has the property that the root has degree two and all other interior vertices have degree three, then ${\mathcal T}$ is {\it binary}. Furthermore, the {\it height} of ${\mathcal T}$ is the number of edges on the longest path from the root to a leaf. If ${\mathcal T}$ is binary, we say ${\mathcal T}$ is {\em balanced} if the size of ${\mathcal T}$ is $2^m$ for some non-negative integer $m$ and the height of ${\mathcal T}$ is $m$.

Let ${\mathcal T}$ be a rooted binary phylogenetic $X$-tree, and let $Y$ be a subset of $X$. Then, the {\it restriction of ${\mathcal T}$ to $Y$}, denoted by ${\mathcal T}|Y$, is the rooted phylogenetic $Y$-tree obtained from the minimal subtree of ${\mathcal T}$ that connects all leaves in $Y$ by suppressing all non-root degree-two vertices. Now let ${\mathcal S}$ be a rooted binary phylogenetic $X'$-tree. If $Y$ is a subset of $X\cap X'$ such that ${\mathcal S}|Y$ and ${\mathcal T}|Y$ are isomorphic, we call ${\mathcal S}|Y$ an {\it agreement subtree} of ${\mathcal S}$ and ${\mathcal T}$. If, amongst all agreement subtrees of ${\mathcal S}$ and ${\mathcal T}$, the restriction ${\mathcal S}|Y$ is of maximum size, then it is called a {\it maximum agreement subtree} (MAST) of ${\mathcal S}$ and ${\mathcal T}$, and $|Y|$ is denoted by $\textrm{mast}({\mathcal S}, {\mathcal T})$.

The first main result of the paper, Theorem~\ref{mainresult}, shows that two balanced rooted binary phylogenetic trees on $n$ leaves do not necessarily have a MAST of size at least $n^{\frac{1}{2}}$.

\begin{theorem}\label{mainresult}
For any $c>0$, there exist balanced rooted binary phylogenetic $X$-trees ${\mathcal S}$ and ${\mathcal T}$, where $n=|X|$, such that ${\rm mast}({\mathcal S}, {\mathcal T}) < cn^{\frac{1}{2}}$.
\end{theorem}

\noindent Theorem~\ref{mainresult} disproves the aforementioned conjecture by Martin and Thatte~\cite[Conjecture 20]{martin2013maximum} that we state next.

\begin{conjecture}[Martin and Thatte~\cite{martin2013maximum}]\label{conjecture}
If ${\mathcal S}$ and ${\mathcal T}$ are two balanced rooted binary phylogenetic $X$-trees, where $n=|X|$, then ${\rm mast}({\mathcal S}, {\mathcal T}) \geq n^{\frac 1 2}$.
\end{conjecture}

{The second main result, Theorem~\ref{thm:lowerbound}, slightly improves the lower bound on the size of a MAST for a pair of balanced rooted binary phylogenetic trees given in \cite{martin2013maximum} from $n^\beta$, where $\beta\sim 0.149$, to $n^{0.17}$.}
\begin{theorem}\label{thm:lowerbound}
{If ${\mathcal S}$ and ${\mathcal T}$ are two balanced rooted binary phylogenetic $X$-trees, where $n=|X|$, then ${\rm mast}({\mathcal S}, {\mathcal T})\ge n^{0.17}>n^{\frac{1}{6}}$.}
\end{theorem}

This paper is organised as follows. Section~\ref{sec:notation} details the notation and definitions used throughout, while Section~\ref{sec:outline} outlines the counterexample construction for when $n=2048$. Section~\ref