I want to draw a tree like this:

The closest I could get to this was the following code:
\documentclass{minimal}
\usepackage{tikz-qtree}
\begin{document}
\begin{tikzpicture}
\tikzset{level 1+/.style={level distance=2\baselineskip}}
\Tree[.IP NP
[.VP [.\node(v){V}; [ \node(v1){}; ] ]
NP
[.PP [.\node(p){P}; \node(p1){}; ]
NP ]
]
]
\draw (v) -- (v1) -- (p1) -- (p);
\end{tikzpicture}
\end{document}
But this involves empty auxiliary nodes and the lines have some space in the middle. Is there a simpler way to do this? Are there options to \drawthat can do this?
