Newton finite differences with Tikz

Newton finite differences with Tikz

I am trying to picture Newton's finite differences method with a Tikz tree. Unfortunately, a few nodes are doubled. Is the approach wrong or is there a solution?
\documentclass{article}
\usepackage{tikz}
\usepackage[active,tightpage]{preview}
\PreviewEnvironment{tikzpicture}
\setlength\PreviewBorder{0pt}%

\tikzset{root/.style={align=center,draw=none},level 2/.style={align=center,left=1.5cm}}

\begin{document}
\begin{tikzpicture}[grow=left,
level 1/.style={sibling distance=15mm},edge from parent/.style={<-,draw},>=latex]

\node[root] {$\delta_{[x_0,x_1,x_2,x_3]}$}
     child {node[level 2] (c1) {$\delta_{[x_0,x_1,x_2]}$}
           child {node[level 2] (c11) {$\delta_{[x_0,x_1]}$}
                 child {node[level 2] (c111) {$\delta_{[x_0]}$}}
                 child {node[level 2] (c112) {$\delta_{[x_1]}$}}
             }
       child {node[level 2] (c12) {$\delta_{[x_1,x_2]}$}
             }
       }
 child {node[level 2] (c2) {$\delta_{[x_1,x_2,x_3]}$}
       child {node[level 2] (c21) {$\delta_{[x_1,x_2]}$}}
       child {node[level 2] (c22) {$\delta_{[x_2,x_3]}$}
             child {node[level 2] (c221) {$\delta_{[x_2]}$}}
             child {node[level 2] (c222) {$\delta_{[x_3]}$}}
             }
       };
\end{tikzpicture}
\end{document}
For instance, nodes c12 and c21 are identical. I want to keep them both without drawing one of them.