2-d trees order its points by a mix of the two coordinates. They have two types of nodes -- x-nodes and y-nodes -- and maintain the following two properties:
- The children of an x-node are y-nodes
- The children of a y-node are x-nodes
Each type of node uses a different comparison to order points.
This causes different levels of the tree to compare points differently, using the following rules:
For every x-node:
- All descendants in the left subtree have a smaller x-coordinate than the point stored at the node. Visually, all descendant points are left of this point.
- All descendants in the right subtree have a larger x-coordinate than the points stored at the node. Visually, all descendant points are right of this point.
For every y-node:
- All descendants in the left subtree have a smaller y-coordinate than the point stored at the node. Visually, all descendant points are below this point.
- All descendants in the right subtree have a larger y-coordinate than the point stored at the node. Visually, all descendant points are above this point.
As it turns out, this property allows us to efficiently perform queries like "which points are within a given radius of a target?"
The following image show querying result in a circle centered at (500,500) with radius 200 from a map of 2000 location-points.
