Binomial trees using forest package

Question

I need to draw a 3-period tree with different elements at each stage, for example:

the upper path:

(12)(12) ---> ()

the lower path:

(13)(12) ---> (123)

how can I draw such a tree?

Edit:

something like this but for a 3-period tree using the forest package to make it more neat:

    \documentclass[12pt, a4paper]{report}
\usepackage{tikz}
\usetikzlibrary{matrix,positioning}
\tikzset{bullet/.style={circle,fill,inner sep=2pt}}


\begin{document}
\begin{tikzpicture}[>=stealth,sloped]
\matrix (tree) [%
      matrix of nodes,
      minimum size=0.1cm,
      column sep=3cm,
      row sep=0.1cm,nodes={text width=8em}
          ]
    {
          &   &  \\
          & {$()$} &   \\
     $(12)$ &   &  \\
          &  {$(123)$} &   \\
          &   &  \\
    };
    \node[bullet,right=0mm of tree-3-1.east](b-3-1){};
    \node[bullet,left=0mm of tree-2-2.west](b-2-2){};
    \node[bullet,left=0mm of tree-4-2.west](b-4-2){};
    \draw[->] (b-3-1) -- (b-2-2) node [midway,above] {$(12)$};
    \draw[->] (b-3-1) -- (b-4-2) node [midway,below] {$(13)$};
  \end{tikzpicture}
\end{document}

Answer 1

Like Alan Munn and cfr I also do not understand the problem. You could make the code you have a bit more versatile by using the auto enumeration that can be found in the forest manual on p. 52, and then all you need to do is to add some basic pgf functions. You may want to adjust the items "cat","mole","duck","marmot",... to your needs.

\documentclass[12pt, a4paper]{report}
\usepackage{forest}
\bracketset{action character=@}
\def\pexp#1#2{\pgfmathsetmacro{\x}{int(#1)}\pgfmathparse{\x>0?(\x>1?"#2^{\x}":"#2"):""}\pgfmathresult}%
\def\btreelabel#1#2{node[midway,sloped,font=\scriptsize,/forest,if n=1{/tikz/below}{/tikz/above}]{$\pexp{#1}{p}\pexp{#2}{(1-p)}$}}%
\def\btree#1#2#3{,grow=east,l*=2,edge label=\btreelabel{#1}{#2},if={#3>0}{append=[@\btree{#1}{#2+1}{#3-1}],append=[@\btree{#1+1}{#2}{#3-1}]}{}}%

\xdef\LstItems{"cat","mole","duck","marmot","koala","rhino","bear","hare","opossum"}
\pgfmathdeclarefunction{mitem}{1}{%
\begingroup%
\pgfmathparse{{\LstItems}[#1-1]}%
\pgfmathsmuggle\pgfmathresult\endgroup%
}%

\begin{document}
\begin{forest}
for tree={grow'=0},tempcounta=1,
before typesetting nodes={for descendants={content/.pgfmath={mitem(int(tempcounta))}, tempcounta+=1}}
[pft@\btree{0}{0}{2}]
\end{forest}
\end{document}

ADDENDUM: Answer to version 2 of your question.

\documentclass[12pt, a4paper]{report}
\usepackage{forest}
\begin{document}
\begin{forest}
for tree={grow'=east,l sep=4em,s sep=3em,circle,inner sep=2pt,fill}
[,label=left:{$(12)$}
 [,label=right:{$()$},edge label={node[midway,sloped,above]{(12)}}]
 [,label=right:{$(123)$},edge label={node[midway,sloped,below]{(13)}}]
]
\end{forest}
\end{document}

LAST ADDENDUM:

\documentclass[12pt, a4paper]{report}
\usepackage{forest}
\begin{document}
\begin{forest}
for tree={grow'=east,l sep=4em,s sep=3em,circle,inner sep=2pt,fill}
[,label=left:{$(12)$}
 [,label=above:{$()$},edge label={node[midway,sloped,above]{$(12)$}}
  [,label=right:$X$,edge label={node[midway,sloped,above]{$x$}}]
  [,label=right:$Y$,edge label={node[midway,sloped,below]{$y$}}]
 ]
 [,label=below:{$(123)$},edge label={node[midway,sloped,below]{(13)}}
  [,label=right:$U$,edge label={node[midway,sloped,above]{$u$}}]
  [,label=right:$V$,edge label={node[midway,sloped,below]{$v$}}]
 ]
]
\end{forest}
\end{document}

Answer 2

Based on answer (which was significantly improved by edit of @marmoth, thank you again!):

\documentclass[12pt, margin=3mm]{standalone}
\usepackage{forest}

\begin{document}
\forestset{
     declare toks={elo}{font=\footnotesize, inner sep=2pt, midway, sloped}, % Edge Label Options
     anchors/.style={anchor=#1,child anchor=#1,parent anchor=#1},
     dot/.style={tikz+={\fill (.child anchor) circle[radius=3pt];}},
     decision edge label/.style 2 args= {
        edge label/.expanded={node[anchor=#1,\forestoption{elo}]{$#2$}}
     },
     decision/.style={if n=1
     {decision edge label={south}{#1}}
     {decision edge label={north}{#1}}
     },
     decision tree/.style={
     for tree={grow'=east,
     s sep=1em,l=13ex,
     if n children=0{anchors=west}{
            if n=1{anchors=south east}{anchors=north east}
                                   },
     },
     anchors=east, 
     dot, for descendants=dot,
     delay={for descendants={split option={content}{;}{content,decision}}},
      }
    }

    \begin{forest} decision tree
[(12)
    [(X);(12),
        [A;a]
        [B;b]
     ]
     [(123);(13),
         [C;c]
         [D;d]
     ]
]
    \end{forest}
\end{document}