Analysis of Recursive and Non-Recursive Complexity with Graph-Based Modeling

Master Theorem with systematic approach and as an essential tool is used to analyses the time complexity of divide and conquer algorithms , it can solve recurrence relation of the theorem of the form T(n)=aT(n/b)+f(n) where a represents the number of subproblems, n/b is the magnitude of each subproblem, and f(n) accounts for the non-recursive work. This paper studies the comparison of growth dynamics of recursive and non-recursive terms in order to classify and determine the asymptotic behaviors of recursive relations. Traditional methods provide the knowledge to an extent but fail in complex scenarios or are proved to be ineffective frequently while applied for solving the recurrence. Master theorem provides a coordinated view for solving the recursive relations by examining the relationship between the parameters, log _b ⁡a and k, where f(n)=Θ(n^k log^p⁡(n) ) , It defines distinct cases for determining whether the recursive or non-recursive work dominates in the overall complexity. This paper examines three main cases with examples and shows where recursion controls, terms raise similarly, or non-recursive work succeeds. Practical examples such as Binary Search, Merge Sort, Strassen's Matrix Multiplication etc are analyzed both through traditional Master Theorem classification and graph-based visual modeling, providing clearer insight into their asymptotic behaviors. This combined approach streamlines the classification process, highlights dominance patterns more intuitively, and offers a new perspective on recurrence analysis in modern computational problems.