Reducing the width of children in a tree

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Here, it is enough to adjust the distance between the children with the level 1 and level 2 style.

level 1/.style={sibling distance=10em},
level 2/.style={sibling distance=5em},

\documentclass[border=5mm]{standalone}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{trees}


\begin{document}
        \begin{tikzpicture}[%sibling distance=10em,
        level 1/.style={sibling distance=10em},
        level 2/.style={sibling distance=5em},
                            every node/.style = {
                                shape=rectangle,
                                rounded corners,
                                draw,
                                align=center,
                                top color=white,
                                bottom color=blue!5}]]
                    \node {$\varnothing$}
                        child{node {F}
                            child{node {$\lnot$p}
                                    child{node {$\lnot$p$\lor$ q}}
                                    child{node {$\lnot$q}}
                                }
                            child{node {p}}
                            }
                        child{node {$\lnot$r}
                                child{node {$\lnot$r$\lor$ s}}
                                child{node {$\lnot$s}}
                            }
                        child{node {r}};
        \end{tikzpicture}
\end{document}