\documentclass[paper=a4, fontsize=11pt]{scrartcl}
\usepackage{forest}

\begin{document}
\begin{forest}
before typesetting nodes = {
    where n children = 0 {
        if content = {\epsilon} {
            append = {
                [{\epsilon},math content,tier=terminal,no edge,color=gray!80]
            }
        }{
            append = {
                [,math content,tier=terminal,edge=dotted,content/.pgfmath=content("!u")]
            }
        }
    }{}
},
for tree={
    math content
}
    [S
        [S_0
            [0]
            [S_0
                [0]
                [S_0
                    [0]
                    [S_0
                        [0]
                        [S_0
                            [0]
                        ]
                    [1]
                    ]
                [1]
                ]
            [1]
            ]
            [1]
        ]
        [S_1
            [2]
            [S_1
                [{\epsilon}]
            ]
        ]
    ]
\end{forest}
\end{document}