Define a ball tree and its applications in nearest neighbor searches.

A ball tree is a binary tree data structure designed for efficient nearest neighbour searches in multi-dimensional spaces.

A ball tree, also known as a metric tree, is a type of binary tree where each node defines a hyper-sphere (or 'ball') containing a subset of the data points. The root of the tree represents the entire dataset, and each child node represents a subset of that data. The division of data is based on a heuristic that aims to minimise the sum of the radii of the child nodes. This structure allows for efficient nearest neighbour searches, as it can quickly eliminate large portions of the data that are outside the search radius.

The process of building a ball tree involves recursively dividing the dataset into two subsets. The division is done in such a way that the sum of the radii of the two resulting balls is minimised. This is typically achieved by choosing a dimension along which to split the data, and then finding a point in that dimension that minimises the sum of the radii. The resulting balls are then used as nodes in the tree, with the parent node containing all the data points in both child nodes.

In terms of applications, ball trees are particularly useful in nearest neighbour searches in multi-dimensional spaces. These are common in many areas of computer science, including computer vision, machine learning, and data mining. For example, in computer vision, nearest neighbour searches can be used to match features in images. In machine learning, they can be used to find similar examples in a training dataset.

The advantage of using a ball tree for these searches is that it can significantly reduce the number of distance calculations needed. This is because the tree structure allows the algorithm to quickly eliminate large portions of the data that are outside the search radius. This makes ball trees an efficient solution for nearest neighbour searches, particularly in high-dimensional spaces where brute-force searches can be very time-consuming.