The objective of this study is to present rigorous proofs for Collatz conjecture and introduce some interesting behavior of the Kaakuma sequence that is a vast generalized form of Collatz sequence. We analyze the behavior of Kaakuma sequence such as scaling up, scaling down, translation, function iteration and uniform growth of inverse tree. In addition to this we investigate relationship of increasing rate, number of iterations of cycles, gap in cycles, and densities of cycles of the Kaakuma sequence and evaluate consistency of tree size density after scaling.Our investigation culminates in the formulation of a set of conjectures encompassing lemmas and postulates, which we rigorously prove using a combination of analytical reasoning, numerical evidence, and exhaustive case analysis. These results provide compelling evidence for the veracity of the Collatz conjecture and contribute to our understanding of the underlying mathematical structure.