Tikz-pictures in prooftrees

Question

I have a problem with the bussproofs package regarding the use of tikz pictures in it. All that I can find on the web seems to assume that one only wants to put simple formulas in the prooftree. But for my application, short graphs are the elements of the prooftree.

This is what I got so far, with the graph abstracted for simplification:

\documentclass[a4paper, 10pt]{article}
\usepackage{tikz}
\usepackage{bussproofs}
\newcommand{\tikzdefault}[1]{\begin{tikzpicture}[shorten >=1pt, node distance=15mm and 12mm, semithick, every text node part/.style={align=center}] #1 \end{tikzpicture}}

\begin{document}
\tikzstyle{node} = [circle, draw, fill=white, text=black, minimum size = 5mm]
\begin{prooftree}
  \AxiomC{}
  \UnaryInfC{\begin{figure}[h]
             \scalebox{0.8}{
             \tikzdefault{
               \begin{scope}
                 \node[node] (v1) [] {$v$};
               \end{scope}
             }
             }
             \end{figure}
  }
\end{prooftree}
\end{document}

I do not have too much experience with LaTeX at all, but I hope someone here has a solution.