Is kd-tree always balanced?

Question

I have used kd-tree algoritham and make tree.

But i found that tree is not balanced so my question is if we used kd-tree algoritham then that tree is always balanced if

Answer 1

It depends on how you build the tree.

If built as originally published, the tree will be balanced, i.e. only at the leaf level it will have at most a height difference of 1. If your data set has 2^n-1 elements, the tree will be perfectly balanced.

When constructed with the median, then half of the objects must be on either branch of the tree, thus it has minimal height and is balanced.

However, this tree cannot be changed then. I am not aware of an insert or remove algorithm that would preserve this property, but YMMV. I bet there are two dozens of kd-tree extensions that aim at rebalancing and making insertions/deletions more effective.

The k-d-tree is not designed for changes, and will quickly lose efficiency. It relies on the median, and thus any change to the tree would worst-case propagate through all of the tree. Therefore, you need to allow some tolerance in the tree quality to support changes. It appears to be a common approach to just keep track of insertions/deletions and rebuild the tree eventually. You cannot combine it with red-black-trees or AVL-trees, because data with more than 1 dimension is not ordered; these trees only work for ordered data. Upon rotation of the tree the splitting axis changes; and there may be elements in either half that suddenly would need to move to the other branch. This does not happen in AVL or red-black trees.

But as you can imagine, people have published several indexes that remain balanced. Such as k-d-b-trees, and R-trees. These also work better for large data that needs to be stored on disk.