Generative recommendation models represent items as sequences of discrete tokens known as semantic IDs. Existing approaches derive these identifiers using residual or hierarchical vector quantization, assuming that tokens carry independent semantic meaning across quantization levels. Revisiting this assumption, we argue that semantic IDs should behave as hierarchical paths where each token's meaning is conditioned on its prefix. This highlights a mismatch between conventional fixed-codebook quantization and the hierarchical structure implicitly learned by generative recommenders. Motivated by this insight, we propose Adaptive Silhouette Tree (AST), a top-down divisive clustering method that constructs semantic IDs as paths within a tree structure. Unlike prior methods with fixed branching factors, AST dynamically determines the number of branches at each node by maximizing the silhouette coefficient, enabling finer partitions in dense regions and compact groupings in homogeneous ones. We further introduce constrained variants that enforce minimum probability mass during splitting to improve supervision for long-tail items. Extensive experiments on five Amazon Review datasets of varying sizes and domains show that AST significantly outperforms strong baselines. These results demonstrate that aligning semantic ID construction with the inherent hierarchical structure of generative recommenders leads to more effective recommendations.

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