Reducing the width of children in a tree

The basic answer to your tree problem is don't use the base TikZ tree methods. Their syntax is clunky and it does no automatic positioning of nodes (as you have discovered.) Instead use forest for making trees. It has a much simpler syntax and does automatic packing of nodes to make trees as compact as possible, but allows you to make them less compact as needed.

But there are many other mistakes in the code you've shown us. It seems like you need to read a good basic introduction of how to use LaTeX more generally. Here are some things I've noticed and fixed (there may be others I've missed.) For a list of common mistakes beginners make see What are the most common mistakes that beginners of (La)TeX and Friends make?. See also What are good learning resources for a LaTeX beginner? and especially the new https://www.learnlatex.org/ site.

1. Numbering things manually. One of the nicest things about LaTeX is its ability to number elements for you. You should never have to number anything manually. So I've replaced your manually numbered paragraphs with subsection commands.
2. If something looks like an itemized list, then make it an itemized list: I've replaced your manual "What will we learn" section with an itemize environment using the enumitem package to format the items.
3. Displayed math is centred by using [ ... ] not begin{center} ... end{center}. And in LaTeX you should never use $$ ... $$ to introduce displayed math. See Why is [ ... ] preferable to $$ ... $$? and also What are the differences between $$, [, align, equation and displaymath? for more options.
4. You should never use to end a paragraph, and in running text you should generally not use it at all. See When to use par and when , newline, or blank lines.

---

documentclass[12pt,letterpaper]{article}
usepackage[margin = 1in]{geometry}
usepackage{amsmath}
usepackage{amssymb}
usepackage{fancyhdr}
usepackage{enumitem}
usepackage[small]{titlesec}
usepackage{tikz}
usepackage[linguistics]{forest}
forestset{mynode/.style={ shape=rectangle,
                                rounded corners,
                                draw,
                                align=center,
                                top color=white,
                                bottom color=blue!5,
                                math content}}


author{emph{Muppuri Siva Jagadesh}}
title{textbf{Discrete Maths}}

pagestyle{fancy}
renewcommand{headrulewidth}{0pt}
renewcommand{footrulewidth}{0pt}

fancyhf{}
rhead{
    Assignment 1
}
rfoot{Page thepage}

begin{document}
    maketitle
    section{LECTURE 1: SAT PROBLEM}
    
    subsection{What will we learn?}
    begin{itemize}[label=$rightarrow$]
        item Satifiability
       item Conjunctive Normal Forms
     end{itemize}   
    subsection{Satisfiability}
        A compound proposition is satisfiable if it is true for at least one truth assignment of the underlying propositional variables.
        [
            text{e.g.: } (p lor lnot q) land (q lor lnot r)
      ]
        
    subsection{}
        begin{enumerate}
            item Replace all $Leftrightarrow$ symbol (if any)
            [
                (p Leftrightarrow q) equiv (p rightarrow q) land (q rightarrow p)
            ]
            item [sum_{n=1}^{infty} 2^{-n} = 1]
        end{enumerate}
        
    
    subsection{}
        begin{forest}for tree={mynode, s sep=1cm}
        [varnothing
            [F
                [lnot p 
                    [lnot p lor q ]
                    [lnot q ]
                ]
                [p ]
            ]
            [lnot r 
                [lnot r lor s ]
                [lnot s ]
            ]
            [r]
        ]
    end{forest}                        
      
    
end{document}