EDIT (Now with 3 solutions)
qtree
+ tikzmark
If you just need to draw arrows later, you can use tikzmark
with qtree
. At least, the following seems to work:
\documentclass{article}
\usepackage{qtree,tikz}
\usetikzlibrary{tikzmark,calc}
\begin{document}
\Tree [ [ [ {\tikzmark{a}$\sqrt{\textsc{Root}}$\tikzmark{b}} {\scshape Voice} ] [.{\scshape Asp} ] ] [.{\scshape {\tikzmark{c}Tns\tikzmark{d}}} ] ]
\begin{tikzpicture}[overlay, remember picture]
\draw [->, shorten >=5pt, shorten <=5pt] ($(pic cs:a)!1/2!(pic cs:b)$) .. controls +(1,-2) and +(-1,-2) .. ($(pic cs:c)!1/2!(pic cs:d)$);
\end{tikzpicture}
\end{document}

forest
I would also use Forest, but would definitely recommend the linguistics
library which provides nice empty nodes
and suitable edges out-of-the-box. To align the terminal nodes, you can add a phantom with zero width but appropriate height.
For example,
\documentclass{standalone}
\usepackage[linguistics]{forest}
\begin{document}
\begin{forest}
nice empty nodes,
for tree={
font=\scshape,
},
before typesetting nodes={
where n children=0{
+content=\makebox[0pt]{\phantom{$\sqrt{X}$}},
}{}
}
[
[
[
[$\sqrt{\textsc{Root}}$]
[Voice]
]
[Asp]
]
[Tns]
]
\end{forest}
\end{document}

tikz-qtree
If you prefer to stick with tikz-qtree
, you should wait to see what Alan Munn can come up with. I've only ever used qtree
and forest
, so the best I can do with tikz-qtree
is

\documentclass[border=10pt]{standalone}
\usepackage{tikz-qtree}
\begin{document}
\newlength\widesttreenodewidth
\settowidth{\widesttreenodewidth}{$\sqrt{\textsc{Root}}$}
\newlength\highesttreenodeht
\settoheight{\highesttreenodeht}{$\sqrt{\textsc{Root}}$}
\newlength\deepesttreenodedpth
\settodepth{\deepesttreenodedpth}{$\sqrt{\textsc{Root}}$}
\tikzset{
every tree node/.append style={text width=\widesttreenodewidth, text depth=\deepesttreenodedpth, text height=\highesttreenodeht, text centered},
execute at begin node=\makebox[0pt]{\phantom{$\sqrt{X}$}},
}
\Tree [ [ [ $\sqrt{\textsc{Root}}$ {\scshape Voice} ] [.{\scshape Asp} ] ] [.{\scshape Tns} ] ]
\end{document}
Note
Forest does take some time to learn and the syntax is a bit different, so it is not something to attempt at the end of a large project, deadline looming, for sure.
forest
'slinguistics
library withnice empty nodes
andsn edges
in most cases.tikz-qtree
rather thanqtree
exactly?tikzmark
withqtree
. I've added a solution based on this to my answer below. I would probably use this if I didn't want to change toforest
and think about usingforest
for future projects, if applicable. That is, unless somebody like Alan Munn comes along with atikz-qtree
solution.