We analyze the ground state phases of two-component (σ≡↑, ↓) population- and mass-balanced (N_↑ = N_↓ and m_↑ = m_↓) but trap-imbalanced (ω_↑ω_↓) fermion mixtures as a function of interaction strength from the weak attraction Bardeen-Cooper-Schrieffer (BCS) to the strong attraction Bose-Einstein condensation (BEC) limit. In the BCS limit, we find that the unpolarized superfluid (UPS) fermions exist away from the central core of the trapping potentials, and are surrounded by partially polarized

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... ormal (PσPN) fermions. As the interactions increase towards unitarity, we find that the central PσPN core first transitions to a UPS, and then expands towards the edges until the entire mixture becomes a UPS in the BEC limit.