Arrow styles in TikZ trees

There are easier ways to draw trees, but if you wish to stick to TikZ's verbosity, you can override the style with another.

\documentclass[border=9pt]{standalone}
\usepackage[]{tikz}

\begin{document}
\tikzset{
  breakarrow/.style={->, dashed},
  myarrow/.style={solid, -},
  varnode/.style = {shape=rectangle, rounded corners,
    draw, align=center,
    top color=red!20, bottom color=white}
}

\begin{tikzpicture}[sibling distance=6em, every node/.style = {shape=rectangle, rounded corners, draw, align=center, top color=white, bottom color=violet!20}]
  \node {/}
  child { node {tmp/}}
  child { node {initrd.img}}
  child { node {usr/}
    child { node {local/}}
    child { node {lib/}}
    child { node [varnode] {$\alpha_{/usr}$} edge from parent [breakarrow]}};
\end{tikzpicture}
\begin{tikzpicture}[sibling distance=6em, every node/.style = {shape=rectangle, rounded corners, draw, align=center, top color=white, bottom color=violet!20}]
  \node [varnode] {$\alpha_{/usr}$}
  child { node {share/} edge from parent [breakarrow]}
  child { node {bin/}  edge from parent [breakarrow]
    child { node {cat} edge from parent [myarrow]}
    child { node [varnode] {$\beta_{/usr/bin}$} edge from parent [breakarrow]}
    child { node {tac} edge from parent [myarrow]}};
\end{tikzpicture}
\begin{tikzpicture}[sibling distance=6em, every node/.style = {shape=rectangle, rounded corners, draw, align=center, top color=white, bottom color=violet!20}]
  \node [varnode] {$\beta_{/usr/bin}$}
  child { node {col} edge from parent [breakarrow]};
\end{tikzpicture}
\end{document}

I would instead do the following:

\documentclass[border=9pt]{standalone}
\usepackage[]{forest}

\begin{document}
\forestset{
  break arrow/.style={edge+={->, dashed}},
  my base node/.style={rounded corners, draw, align=center,},
  var node/.style = {my base node, top color=red!20, bottom color=white},
  my node/.style={my base node, top color=white, bottom color=violet!20,},
}
\begin{forest}
  for tree={my node, s sep'=3em, l sep'+=1em, }
  [$\alpha_{/usr}$, var node
    [share/]
    [bin/, break arrow
      [cat]
      [$\beta_{/usr/bin}$, var node, break arrow]
      [tac]
    ]
  ]
\end{forest}
\end{document}

Then you don't have to override anything and can apply styles precisely to the edges and nodes you want.