Pair Hidden Markov Model (Pair-HMM) forward algorithm is gaining increasing popularity in biological research tools. We propose a novel non-cooperative structure of Pair-HMM forward algorithm accelerator on Field Programmable Gate Array (FPGA). We employ a task-level parallel scheme in the structure. We design the non-cooperative Processing Element (PE) to complete Pair-HMM forward algorithm independently. Our three-layer tree topology improves the scalability for different FPGAs. Compared to previous works, our structure reduces the idle cycles which occurs in the systolic array structure and the PE ring structure. Compared with the PE ring, our implementation on Arria 10 can achieve 1.19× speedups.