Flash memories are in ubiquitous use for storage on sensor nodes, mobile devices, and enterprise servers. However, they present significant challenges in designing tree indexes due to their fundamentally different read and write characteristics in comparison to magnetic disks. In this paper, we present the Lazy-Adaptive Tree (LA-Tree), a novel index structure that is designed to improve performance by minimizing accesses to ash. The LA-tree has three key features: 1) it amortizes the cost of node reads and writes by performing update operations in a lazy manner using cascaded buffers, 2) it dynamically adapts buffer sizes to workload using an online algorithm, which we prove to be optimal under the cost model for raw NAND ashes, and 3) it optimizes index parameters, memory management, and storage reclamation to address ash constraints. Our performance results on raw NAND ashes show that the LA-Tree achieves 2x to 12x gains over the best of alternate schemes across a range of workloads and memory constraints. Initial results on SSDs are also promising, with 3x to 6x gains in most cases.