Let’s invent B(+)-trees.
Say we have some things, and we want to be able to store and look them up in order. We could just put them in a big sorted array:
This is great, but it’s hard to mutate – to insert or delete an element, we might have to move all the others!
So, instead, let’s split our array into fixed-size blocks – which can be in any order – and keep references to the blocks in sorted order:
Each block is allowed to have some empty space, which we can use for insertions.
If we inserted 55 above, we’d get:
So we only need to move up to a blocksworth of elements to insert into a block. What if the block we want to insert into is full? We can split a block into two halves. Say we want to insert 15 above:
[10,20,30,40] [50,55,60,70] [10,20,--,--] [30,40,--,--] [50,55,60,70] [10,15,20,--] [30,40,--,--] [50,55,60,70]
When we split a full block, it becomes half-empty. In order not to have too many blocks, we don’t permit blocks to be any emptier than that, which means at most 50% of block space is wasted (but see below).
(Deletion is mostly like insertion: If, after deleting an element, a block is less than half-full, we either merge it with its neighbor if they’re both half-full, or we move an element over from its neighbor. If it has no neighbors, we just leave it as it is.)
The next question is: How do we store the sorted list of blocks, which there might be a lot of? We can just use the same structure: Blocks indexing blocks indexing values (until the top-level list of blocks fits in a single block). This is just a tree structure. The layer of blocks containing actual data is the same; the layers above it store pointers to the next layer.
Since each block is a fixed size, the depth of the tree must be variable: Some number of layers of blocks indexing blocks indexing data. To insert into a full block, we split it in two, and insert the new node into its parent, and so on up to the root block. When we split the root we create a new root pointing to the two new nodes (which is how the tree grows).
(The root is a bit special: If it contains data, it can have as few as zero items, since we might just not have data. If it contains other blocks, it can have as few as two items; if it ever only has one child, you can replace it with its own child, which is how the tree shrinks.)
The above is skipping over an important detail: We don’t just want blocks to be ordered, we want to be able to look our things up by key. So for each block we want to know the range of keys it can contain. If a block contains keys, that’s easy – the range is [first,last]. If a block contains blocks, we could store the full range of each subblock –
(l1,r1) (l2,r2) (l3,r3) (l4,r4) [ b1, b2, b3, b4 ]
– but we really just care about finding the right block, so l1 and r4 are irrelevant, and r1/l2,r2/l3,r3/l4 are redundant – any value in that range would work, since it just tells us which side to descend into. So we just need to store n-1 keys:
l2 l3 l4 [ b1, b2, b3, b4 ]
This got linked more widely than I expected, so I’ll add a few notes:
The main distinction between a B-tree and a B+-tree is that in B+ trees, data is stored only in leaves (the last layer), and the keys in the other layers are redundant copies. In classic B-trees, the internal layers have their own keys and values. There are many small variations you can make in this family of data structures, but the main idea is the same.
B-trees are particularly suited for on-disk storage, which is what they were originally designed for, but they’re also good for in-memory storage. There’s rarely a reason to use binary trees, with nodes that don’t even fill a cache line, nowadays.
Deletion as described above is deceptively complicated – by far the trickiest to implement correctly. One way to simplify it is “relaxed deletion” – just let nodes get underfull, and only delete them when they’re completely empty. The theoretical guarantees of this aren’t as good, but in practice it works well (sometimes even better than “proper” deletion), and the total tree size is still bounded by the total number of inserts.
An interesting observation is that this is one of the most efficient data structures for ordered key-value maps; but up to the block size of, say, 32 items, a B-tree is just a sorted array, probably with linear search! For only a small number of items, an array is optimal in practice. For a medium number of items, a two-level tree (just an array of blocks) is simpler than a full B-tree and works well. A regular B-tree naturally turns into one of these at low sizes, of course.