We establish sharp extremal bounds on the Albertson and Sigma irregularity indices for trees with prescribed degree sequences, with emphasis on caterpillar trees as key extremal configurations. New lower and upper bounds are derived in terms of maximum degree, average degree, and auxiliary sequence parameters, highlighting the quadratic growth of the Sigma index relative to the linear Albertson index. Closed-form expressions, direct index relations, and empirical validation confirm the bounds' tightness. These findings extend prior work on linear irregularity measures and offer precise tools for analyzing degree-heterogeneous trees in graph theory and chemical graph applications.