A network architecture for tree-like passive optical networks (PONs) with high splitting ratios is proposed and experimentally demonstrated. Splits >1000 are shown to be feasible thanks to a noise-powered amplifier that is built-in inside the power splitter. The proposed noise pumping technique recycles the natural amplified spontaneous emission (ASE) of a semiconductor optical amplifier at the customer premises, avoiding the use of an electrical power supply in the fiber plant. The impact of the location of this noise-powered extender box inside the signal distribution element of the tree is analyzed in terms of obtained signal gain, ASE accumulation among multiple extender boxes, and transmission performance. Finally, full-duplex transmission at 10 Gb/s with burst-mode upstream is demonstrated in a particular scenario with a split of 1:1000 and a loss budget of 39.5 dB, revealing that power margins >3 of dB can be provided.