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\title{Dynamic Branching Adjustment Complexity Proof}
\author{Shyamal Suhana Chandra}
\date{Copyright (C) 2025}

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\section{Theorem: Dynamic Branching Adjustment Complexity}

\textbf{Statement:} Adjusting branching factor in an N-way splay tree takes $O(1)$ amortized time per operation.

\section{Proof}

\begin{enumerate}
    \item Branching adjustment triggered when: $|\text{children}| > \text{maxChildren}$ or optimal branching changes
    \item Split operation: $O(t)$ where $t$ is current branching factor
    \item Since $t \leq \sqrt{n}$ (by design), split cost: $O(\sqrt{n})$
    \item Split occurs at most once per $O(\sqrt{n})$ insertions
    \item Amortized cost: $\frac{O(\sqrt{n})}{O(\sqrt{n})} = O(1)$
\end{enumerate}

\textbf{Conclusion:} Dynamic branching adjustment has $O(1)$ amortized time complexity.

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