Vertical line in middle of a binomial tree (lattice)

Question

This is a really tough problem that I have no idea how to solve. I wish to have a dotted veritcal line in the middle of my binomial tree, with a $\tau$ underneath.. The kind of thing I want is as follows:

The code for the image WITHOUT this line (or the associated \tau) is as follows:

 \documentclass{article}

 \usepackage{tikz}
 \usetikzlibrary{matrix,fit,backgrounds}



  \begin{document}


 \begin{tikzpicture}[>=stealth,sloped]
    \matrix (tree) [%
      matrix of nodes,
      minimum size=1cm,
      column sep=2.5cm,
      row sep=0.4cm,
    ]
    {  
              &                    &                     &  $_\tau S_0$             \\
              &                    &  $_{2\Delta t}S_0$  &                  \\
              & $_{\Delta t}S_0$   &                     &  $_\tau S_1$             \\
      $_0S_0$ &                    &  $_{2\Delta t}S_1$  &                  \\
              & $_{\Delta t}S_1$   &                      &  $_\tau S_2$         \\
              &                    &  $_{2\Delta t}S_2$  &                \\
              &                    &                     &  $_\tau S_3$    \\
    };

    \draw[->] (tree-4-1) -- (tree-3-2) node [midway,above] {$p$};
    \draw[->] (tree-4-1) -- (tree-5-2) node [midway,below] {$1-p$};
    \draw[->] (tree-3-2) -- (tree-2-3) node [midway,above] {};
    \draw[->] (tree-3-2) -- (tree-4-3) node [midway,below] {};
    \draw[->] (tree-5-2) -- (tree-4-3) node [midway,above] {};
    \draw[->] (tree-5-2) -- (tree-6-3) node [midway,below] {};

    \draw[->] (tree-2-3) -- (tree-1-4) node [midway,above] {};
    \draw[->] (tree-2-3) -- (tree-3-4) node [midway,below] {};
    \draw[->] (tree-4-3) -- (tree-3-4) node [midway,above] {};
    \draw[->] (tree-4-3) -- (tree-5-4) node [midway,below] {};
    \draw[->] (tree-6-3) -- (tree-5-4) node [midway,above] {};
    \draw[->] (tree-6-3) -- (tree-7-4) node [midway,below] {};

  \end{tikzpicture}

  \end{document}

Answer 1

The calc library makes this an easy task, but you have to assign names to empty matrix cells:

\usetikzlibrary{calc}
\begin{tikzpicture}[>=stealth,sloped]
    \matrix (tree) [%
      matrix of nodes,
      minimum size=1cm,
      column sep=2.5cm,
      row sep=0.4cm,
    ]
    {  
              & |(top-left)|                   & |(top-right)|                     &  $_\tau S_0$             \\
              &                    &  $_{2\Delta t}S_0$  &                  \\
              & $_{\Delta t}S_0$   &                     &  $_\tau S_1$             \\
      $_0S_0$ &                    &  $_{2\Delta t}S_1$  &                  \\
              & $_{\Delta t}S_1$   &                      &  $_\tau S_2$         \\
              &                    &  $_{2\Delta t}S_2$  &                \\
              & |(bottom-left)|                   & |(bottom-right)|                    &  $_\tau S_3$    \\
    };

    \draw[->] (tree-4-1) -- (tree-3-2) node [midway,above] {$p$};
    \draw[->] (tree-4-1) -- (tree-5-2) node [midway,below] {$1-p$};
    \draw[->] (tree-3-2) -- (tree-2-3) node [midway,above] {};
    \draw[->] (tree-3-2) -- (tree-4-3) node [midway,below] {};
    \draw[->] (tree-5-2) -- (tree-4-3) node [midway,above] {};
    \draw[->] (tree-5-2) -- (tree-6-3) node [midway,below] {};

    \draw[->] (tree-2-3) -- (tree-1-4) node [midway,above] {};
    \draw[->] (tree-2-3) -- (tree-3-4) node [midway,below] {};
    \draw[->] (tree-4-3) -- (tree-3-4) node [midway,above] {};
    \draw[->] (tree-4-3) -- (tree-5-4) node [midway,below] {};
    \draw[->] (tree-6-3) -- (tree-5-4) node [midway,above] {};
    \draw[->] (tree-6-3) -- (tree-7-4) node [midway,below] {};

    % τ-line
    \draw[dashed] ($(top-left.north)!.5!(top-right.north)$) -- ($(bottom-left.south)!.5!(bottom-right.south)$);
    \node at ($(bottom-left.south)!.5!(bottom-right.south)-(0,1em)$) {$\tau$};

  \end{tikzpicture}

Answer 2

You can use orthogonal intersections via |- and -| coordinate qualifiers.

\documentclass{article}

\usepackage{tikz}
\usetikzlibrary{matrix,fit,backgrounds}



\begin{document}


 \begin{tikzpicture}[>=stealth,sloped]
    \matrix (tree) [%
      matrix of nodes,
      minimum size=1cm,
      column sep=2.5cm,
      row sep=0.4cm,
    ]
    {  
              &                    &                     &  $_\tau S_0$             \\
              &                    &  $_{2\Delta t}S_0$  &                  \\
              & $_{\Delta t}S_0$   &                     &  $_\tau S_1$             \\
      $_0S_0$ &                    &  $_{2\Delta t}S_1$  &                  \\
              & $_{\Delta t}S_1$   &                      &  $_\tau S_2$         \\
              &                    &  $_{2\Delta t}S_2$  &                \\
              &                    &                     &  $_\tau S_3$    \\
    };

    \draw[->] (tree-4-1) -- (tree-3-2) node [midway,above] {$p$};
    \draw[->] (tree-4-1) -- (tree-5-2) node [midway,below] {$1-p$};
    \draw[->] (tree-3-2) -- (tree-2-3) node [midway,above] (a) {};
    \draw[->] (tree-3-2) -- (tree-4-3) node [midway,below] {};
    \draw[->] (tree-5-2) -- (tree-4-3) node [midway,above] {};
    \draw[->] (tree-5-2) -- (tree-6-3) node [midway,below] {};

    \draw[->] (tree-2-3) -- (tree-1-4) node [midway,above] {};
    \draw[->] (tree-2-3) -- (tree-3-4) node [midway,below] {};
    \draw[->] (tree-4-3) -- (tree-3-4) node [midway,above] {};
    \draw[->] (tree-4-3) -- (tree-5-4) node [midway,below] {};
    \draw[->] (tree-6-3) -- (tree-5-4) node [midway,above] {};
    \draw[->] (tree-6-3) -- (tree-7-4) node [midway,below] {};

\draw[dashed] (tree-7-4 -| a) -- (tree-1-4 -| a);
  \end{tikzpicture}

  \end{document}