Draw a cross symbol through arrow in tikzpicture

\documentclass[12pt]{article}
\usepackage[utf8]{inputenc}
   \usepackage{tikz-qtree-compat}
 \tikzset{every tree node/.style={baseline=(top.base),
level distance=2em, sibling distance=4em, align=center,
parent anchor=south, child anchor=north, anchor=north}, sibling distance=15pt}
\usetikzlibrary{positioning}

\tikzset{pics/.cd,
    % Marque croix en diagonale
    Cross/.style args={#1 and #2}{%
        code = {%
        \draw[#2,rotate=45,scale=1.4,very thick]
                (0,#1 pt) -- (0,-#1 pt) ;
        \draw[#2,rotate=-45,scale=1.4,very thick]
                (0,#1 pt) -- (0,-#1 pt) ;
        }
    },
    Cross/.default={2.5 and gray!25!black}}


  \begin{document}

\begin{center}
\begin{tikzpicture}  [inv/.style={overlay, coordinate  }, sibling distance=10pt] 
\Tree [.XP  \node(AA){X};               [.YP [.Y   ] [.ZP Z  \node(BB){X}; ]   ]]]]
  \draw[ ->  ] (AA.south)
.. controls +(south:5em)
 and +(south:2em) .. (BB.south) pic[pos=.35,-,rotate=30] {Cross={4 and red}} ;



\end{tikzpicture}
  \end{center}
 \end{document}

Some easy solutions:

\documentclass[12pt]{article}
\usepackage[utf8]{inputenc}
   \usepackage{tikz-qtree-compat}
 \tikzset{every tree node/.style={baseline=(top.base),
level distance=2em, sibling distance=4em, align=center,
parent anchor=south, child anchor=north, anchor=north}, sibling distance=15pt}
 \usetikzlibrary{positioning}
 \usetikzlibrary{decorations}



  \begin{document}

\begin{center}
\begin{tikzpicture}  [inv/.style={overlay, coordinate  }, sibling distance=10pt] 
\Tree [.XP  \node(AA){X};               [.YP [.Y   ] [.ZP Z  \node(BB){X}; ]   ]]]]
  \draw[ ->,red] (AA.south)
.. controls +(south:5em)
  and +(south:2em) .. (BB.south) node[pos=0.3,sloped]{\Large$X$};
    \draw[ -> ,blue] ([xshift=-15,yshift=-15]AA.south)
.. controls +(south:5em)
 and +(south:2em) .. ([xshift=-15,yshift=-15]BB.south) node[pos=0.3,sloped,rotate=45]{\Large$|$}node[pos=0.3,rotate=-65]{\Large$|$};
\end{tikzpicture}
  \end{center}
 \end{document}

PS: Shifted just to demonstrate next to each other

And yet another proposal, based on this answer, mainly to say that Ignasi did this some time ago.

\documentclass[12pt]{article}
\usepackage[utf8]{inputenc}
   \usepackage{tikz-qtree-compat}
   \usetikzlibrary{shapes.misc}

   \usetikzlibrary{decorations.markings}
 \tikzset{every tree node/.style={baseline=(top.base),
level distance=2em, sibling distance=4em, align=center,
parent anchor=south, child anchor=north, anchor=north}, sibling distance=15pt}
\usetikzlibrary{positioning}


% from https://tex.stackexchange.com/a/124064/121799
\tikzset{cross/.style={cross out, draw, 
         minimum size=2*(#1-\pgflinewidth), 
         inner sep=0pt, outer sep=0pt}}


  \begin{document}

\begin{center}
\begin{tikzpicture}  [inv/.style={overlay, coordinate  }, sibling distance=10pt] 
\Tree [.XP  \node(AA){X};               [.YP [.Y   ] [.ZP Z  \node(BB){X}; ]   ]]]]
  \draw[ ->] (AA.south)
.. controls +(south:5em)
 and +(south:2em) .. (BB.south) node[pos=0.4,cross=5pt,sloped]{};
\end{tikzpicture}
  \end{center}
 \end{document}