Note that I think I ought not answer do-it-for-me questions such as this one. When I do so, I do so for me. I provide code for the heck of it. I generally do not explain it as I have no idea what to explain and, besides, I spent my efforts setting the puzzle up, in addition to solving it. I am typically less than sympathetic to requests for changes, explanations, tweaks and adjustments. These are left as exercises for the under-exercised reader. If my code happens to be useful, so be it; if not, tough. Good answers are responses to good questions. Do-it-for-mes cannot expect the same privileges.
Caveat emptor.
Yes, of course it uses TikZ. It is a tree. Every tree needs a Forest.
\documentclass[border=10pt]{standalone}
\usepackage{amssymb}
\usepackage[edges]{forest}
\newcommand*{\lif}{\ensuremath{\mathbin{\rightarrow}}}
\begin{document}
\forestset{
declare boolean={T}{true},
F/.style={not T},
lr tableau/.style={
before typesetting nodes={
for tree={
math content,
fork sep'=7.5pt,
if T={
parent anchor=east,
child anchor=east,
anchor=mid east,
}{
parent anchor=west,
child anchor=west,
anchor=mid west,
},
},
},
forked edges,
before computing xy={
where level=0{if T={parent anchor=north east}{parent anchor=north west}}{
if={
>O_< {!u.n children}{2}%
}{l'=\baselineskip}{},
},
where n children=0{if T={child anchor=south east}{child anchor=south west}}{},
}
},
close/.style={
if n=1{label={[label distance=0pt, anchor=north]-135:\textsf{x}}}{label={[label distance=0pt, anchor=north]-45:\textsf{x}}},
},
}
\begin{forest}
lr tableau,
[\lnot \phi, label=left:\checkmark
[\alpha \lif \beta, label=left:\checkmark
[\beta \lif \phi, label=left:\checkmark
[\lnot\alpha, F, label=right:\checkmark
[\alpha
[\phi, F
[\alpha, F, close ]
[\beta
[\beta, F, close ]
[\phi, close ]
]
]
]
]
]
]
]
\end{forest}
\end{document}

forestortikz-tree. For aforestsolution check this question: https://tex.stackexchange.com/questions/289642/how-to-draw-a-proper-decision-tree Or useprooftree: https://tex.stackexchange.com/questions/289268/how-do-i-concisely-typeset-propositional-proof-trees – jaytar Sep 22 '17 at 19:03prooftreesand notprooftree. If you really want to avoid TikZ, all of these options are out anyway. In that case, the only tree-drawing packages I know of arepstricksorqtree. I wouldn't try to useqtreefor this. I don't know much aboutpstricks, but I'm sure you can use it if you wish. Like TikZ, it is powerful and flexible. – cfr Sep 22 '17 at 22:40