An example of how to use the bussproofs.sty file for setting out proof trees.

prooftrees.tex

% I'm assuming you have the package imported already
\usepackage{bussproofs}

% In these examples, I like indenting the InfC part of each proof tree from the AxiomC(s) that spawned it,
% and keeping those AxiomC(s) on the same indent level, but I keep the bottom, final InfC unindented
% This is a personal choice but it makes it easier to keep track of what's happening for large trees

% A rule with no assumptions (i.e. an axiom)
\begin{prooftree}
  \AxiomC{}
    \UnaryInfC{$A$ \textbf{Foo}}
\end{prooftree}

% A rule with one assumption
\begin{prooftree}
  \AxiomC{$A$ \textbf{Foo}}
    \UnaryInfC{$f(A)$ \textbf{Foo}}
\end{prooftree}

% A rule with two assumptions
% Note how UnaryInfC changed to BinaryInfC; read the docs for more info
\begin{prooftree}
  \AxiomC{$A_1$ \textbf{Foo}}
  \AxiomC{$A_2$ \textbf{Foo}}
    \BinaryInfC{$f(A_1) + f(A_2)$ \textbf{Foo}}
\end{prooftree}

% A rule with the top assumptions being proof trees themselves
% You can keep doing this for as long as you like, just be careful with the LaTeX
\begin{prooftree}
  \AxiomC{}
    \UnaryInfC{$A_1$ \textbf{Foo}}
  \AxiomC{}
    \UnaryInfC{$A_2$ \textbf{Foo}}
  \BinaryInfC{$f(A_1) + f(A_2)$ \textbf{Foo}}
\end{prooftree}

% Naming a rule with a label on the right
\begin{prooftree}
  \AxiomC{}
    \RightLabel{$R$}
    \UnaryInfC{$A$ \textbf{Foo}}
\end{prooftree}

% Aligning multiple proof trees
% If you're attempting to fit too many trees on a line it will break them apart, so be careful
\begin{prooftree}
  \AxiomC{}
    \UnaryInfC{$A$ \textbf{Foo}}
  \DisplayProof \hskip 1em
  \AxiomC{}
    \UnaryInfC{$B$ \textbf{Foo}}
  \DisplayProof \hskip 1em
  \AxiomC{}
    \UnaryInfC{$C$ \textbf{Foo}}
\end{prooftree}

% Aligning multiple proof trees correctly which are 'vertically uneven' using vphantom
\begin{prooftree}
  \AxiomC{$A_1$ \textbf{Foo}}
  \AxiomC{$A_2$ \textbf{Foo}}
    \BinaryInfC{$f(A_1) + f(A_2)$ \textbf{Foo}}
  \DisplayProof \hskip 1em  
  % This proof tree here contains no assumption so the final inference line will be uneven
  % with the previous tree's final inference line; so we place the content of an assumption
  % in a \vphantom environment which creates a zero-width box of the same height as what
  % you pass in, in this case the rule from that first tree
  \AxiomC{\vphantom{$A_1$ \textbf{Foo}}}
    \UnaryInfC{$A$ \textbf{Foo}}
\end{prooftree}