Algolia, a data storage company, uses tree search to efficiently manage large amounts of input data. Tree search is an algorithm that works by creating a branching structure where each node represents a decision point based on query relevancy. This technique allows for quick traversal through the tree and has been used in various applications such as JPEG and MP3 compression, contact list searches, and programming language compilers. As the dataset grows, the search speed and storage space of Algolia's indexes become more efficient, making it possible to handle massive datasets like those found on GitLab, King Arthur Baking, GoFundMe, and Gucci product databases.