Typeset trees using TikZ/PGF

Some combinatorial trees typeset using TikZ/PGF. The Linux filesystem hierarchy:

\begin{tikzpicture}
[-,thick]
\footnotesize
\node {\texttt{/}} [edge from parent fork down]
  child {node {\texttt{bin}}}
  child {node {\texttt{etc}}}
  child {node {\texttt{home}}
    child {node {\texttt{anne}}}
    child {node {\texttt{sam}}}
    child {node {$\dots$}}
  }
  child {node {\texttt{lib}}}
  child {node {\texttt{opt}}}
  child {node {\texttt{proc}}}
  child {node {\texttt{tmp}}}
  child {node {\texttt{usr}}
    child {node {\texttt{bin}}
      child {node {\texttt{acyclic}}}
      child {node {\texttt{diff}}}
      child {node {\texttt{dot}}}
      child {node {\texttt{gc}}}
      child {node {\texttt{neato}}}
      child {node {$\dots$}}
    }
    child {node {\texttt{include}}}
    child {node {\texttt{local}}}
    child {node {\texttt{share}}}
    child {node {\texttt{src}}}
    child {node {$\dots$}}
  }
  child {node {$\dots$}};
\end{tikzpicture}

Classification tree of organisms:

\begin{tikzpicture}
[sibling distance=6cm,-,thick]
\footnotesize
\node {organism}
  child {node {plant}
    [sibling distance=2cm]
    child {node {tree}
      child {node {deciduous}}
      child {node {evergreen}}
    }
    child {node {flower}}
  }
  child {node {animal}
    [sibling distance=2.5cm]
    child {node {invertebrate}}
    child {node {vetebrate}
      [sibling distance=4.7cm]
      child {node {bird}
        [sibling distance=1.5cm]
        child {node {finch}}
        child {node {rosella}}
        child {node {sparrow}}
      }
      child {node {mammal}
        [sibling distance=1.5cm]
        child {node {dolphin}}
        child {node {human}}
        child {node {whale}}
      }
    }
  };
\end{tikzpicture}

The Bernoulli family tree of mathematicians:

\begin{tikzpicture}
[-,thick,%
  every node/.style={shape=rectangle,inner sep=3pt,draw,thick}]
\footnotesize
\node {Nikolaus senior} [edge from parent fork down]
  [sibling distance=4cm]
  child {node {Jacob}}
  child {node {Nicolaus}
    child {node {Nicolaus I}}
  }
  child {node {Johann}
    [sibling distance=2cm]
    child {node {Nicolaus II}}
    child {node {Daniel}}
    child {node {Johann II}
      child {node {Johann III}}
      child {node {Daniel II}}
      child {node {Jakob II}}
    }
  };
\end{tikzpicture}

An expression tree:

\begin{tikzpicture}
[-,thick]
\node {$+$}
  [sibling distance=2.5cm]
  child {node {$\times$}
    [sibling distance=1cm]
    child {node {$a$}}
    child {node {$a$}}
  }
  child {node {$\times$}
    [sibling distance=1cm]
    child {node {$2$}}
    child {node {$a$}}
    child {node {$b$}}
  }
  child {node {$\times$}
    [sibling distance=1cm]
    child {node {$b$}}
    child {node {$b$}}
  };
\end{tikzpicture}

Simple graphs, bridges of Konigsberg and directed graphs

Some combinatorial graphs drawn using TikZ/PGF. The seven bridges of Konigsberg:

\begin{tikzpicture}
[lineDecorate/.style={-,thick},%                                                
  nodeDecorate/.style={shape=circle,inner sep=2pt,draw,thick}]
%% nodes or vertices                                                            
\foreach \nodename/\x/\y/\direction/\navigate in {
  a/0/0/left/west, b/0/2/left/west, c/0/4/left/west, d/4/2/right/east}
{
  \node (\nodename) at (\x,\y) [nodeDecorate] {};
  \node [\direction] at (\nodename.\navigate) {\footnotesize$\nodename$};
}
%% edges or lines                                                               
\path
\foreach \startnode/\endnode in {a/d, b/d, c/d}
{
  (\startnode) edge[lineDecorate] node {} (\endnode)
}
\foreach \startnode/\endnode in {a/b, b/c, c/b, b/a}
{
  (\startnode) edge[lineDecorate,bend left] node {} (\endnode)
};
\end{tikzpicture}

A house graph:

\begin{tikzpicture}
[lineDecorate/.style={-,thick},%                                                
  nodeDecorate/.style={shape=circle,inner sep=2pt,draw,thick}]
%% nodes or vertices                                                            
\foreach \nodename/\x/\y/\direction/\navigate in {
  a/0/5/above/north, b/2/3/right/east, e/-2/3/left/west,
  c/2/0/right/east, d/-2/0/left/west}
{
  \node (\nodename) at (\x,\y) [nodeDecorate] {};
  \node [\direction] at (\nodename.\navigate) {\footnotesize$\nodename$};
}
%% edges or lines                                                               
\path
\foreach \startnode/\endnode in {a/b, b/c, b/e, c/d, d/e, e/a}
{
  (\startnode) edge[lineDecorate] node {} (\endnode)
};
\end{tikzpicture}

A weighted multigraph:

\begin{tikzpicture}
[nodeDecorate/.style={shape=circle,inner sep=1pt,draw,thick}]
%% nodes or vertices                                                            
\foreach \nodename/\x/\y in {v_1/4/0, v_2/0/0, v_3/0/3, v_4/4/3, v_5/7/1.5}
{
  \node (\nodename) at (\x,\y) [nodeDecorate] {\scriptsize$\nodename$};
}
%% edges or lines                                                               
\tikzstyle{EdgeStyle}=[->,>=stealth,thick]
\tikzstyle{LabelStyle}=[fill=white]
\foreach \startnode/\endnode/\bend/\weight in {
  v_1/v_2/bend left/1, v_1/v_3/bend left/3, v_2/v_3/bend left/1,
  v_2/v_4/bend left/3, v_3/v_1/bend left/1, v_3/v_2/bend left=0/2,
  v_3/v_4/bend left/1, v_4/v_1/bend left=0/3, v_4/v_5/bend left/2,
  v_5/v_1/bend left=0/3, v_5/v_1/bend left/6, v_5/v_4/bend left/1}
{
  \scriptsize
  \Edge[label=$\weight$,style=\bend](\startnode)(\endnode)
}
\end{tikzpicture}

QED symbol for end of proof

The amssymb package defines a default Halmos symbol that automatically appears when you use the proof environment. That default Halmos symbol is . But if you want that to be filled with black, you could redefine the Halmos symbol. Put the following in your preamble:

\renewcommand{\qed}{\hfill \mbox{\raggedright \rule{0.1in}{0.1in}}}

Better still, you could customize what symbol you want to use as your QED symbol. Just replace the macro \rule{0.1in}{0.1in} with the the symbol of your choice.