We present AlGaN/GaN nanostructured Schottky barrier diodes (SBDs) on silicon substrate with high breakdown voltage (V-BR) and low reverse leakage current (I-R), based on a hybrid of tri-anode and tri-gate architectures. The fabricated SBDs presented a small turn-ON voltage (V-ON) of 0.76 +/- 0.05 V, since the tri-anode architecture formed direct Schottky contact to the 2-D electron gas (2DEG). The reverse characteristic was controlled electrostatically by an embedded tri-gate transistor, instead of relying only on the Schottky barrier. This resulted in low I-R below 10 and 100 nA/mm at large reverse biases up to 500 and 700 V, respectively. In addition, these devices exhibited record V-BR up to 1325 V at I-R of 1 mu A/mm, rendering an excellent high-power figure-of-merit (FOM) of 939 MW/cm(2) and demonstrating the significant potential of nanostructured GaN SBDs for future efficient power conversion.