Conventional vision backbones, despite their success, often construct features through a largely uniform cascade of operations, offering limited explicit pathways for adaptive, iterative refinement. This raises a compelling question: can principles from classical search algorithms instill a more algorithmic, structured, and logical processing flow within these networks, leading to representations built through more interpretable, perhaps reasoning-like decision processes? We introduce DeepTraverse, a novel vision architecture directly inspired by algorithmic search strategies, enabling it to learn features through a process of systematic elucidation and adaptive refinement distinct from conventional approaches. DeepTraverse operationalizes this via two key synergistic components: recursive exploration modules that methodically deepen feature analysis along promising representational paths with parameter sharing for efficiency, and adaptive calibration modules that dynamically adjust feature salience based on evolving global context. The resulting algorithmic interplay allows DeepTraverse to intelligently construct and refine feature patterns. Comprehensive evaluations across a diverse suite of image classification benchmarks show that DeepTraverse achieves highly competitive classification accuracy and robust feature discrimination, often outperforming conventional models with similar or larger parameter counts. Our work demonstrates that integrating such algorithmic priors provides a principled and effective strategy for building more efficient, performant, and structured vision backbones.