I create a tree by node/child as follows:
\begin{tikzpicture}
\tikzstyle{nodestyle} = [circle,draw,minimum size=5pt]
\node [nodestyle, inner sep=0pt] {$\{X\}|_{\aleph_0}$}
child {node {$\vdots$}
child {node [nodestyle, inner sep=4pt] (a) {$X$}}
};
\path (0,0.7) node[above] {$X,X_1,X_2\in\mathcal{E}$};
\begin{scope}[xshift=2.7cm]
\node [nodestyle, inner sep=0pt] {$\{S\}|_{\aleph_0}$}
child {node {$\vdots$}
child {node [nodestyle, inner sep=3pt] (a) {$S$}
child {node [nodestyle,inner sep=0pt] {$\{X_1\}$}
child {node [nodestyle, inner sep=1pt] {$X_1$}}
}
child {node [nodestyle,inner sep=1pt] (e) {$X_2$}}
}
};
\path (0,0.7) node[above] {$S=\{\{X_1\},X_2\}$};
\end{scope}
\end{tikzpicture}
I want to set distance from branch above cdots to be same as that of below cdots. Thanks.

minimum size=1cmto1cmto some smaller value and/or play with the line style, e.g.loosely dottedinstead ofdotted. – Feb 02 '18 at 18:27