B-Tree - Why can't there be a node with an even number of keys?

Question

I'm trying to implement a B-Tree according to the chapter "B-Trees" in "Introduction to Algorithms".

What I don't quite get is the "minimal degree". In the book it is stated that the degree is a number which expresses the lower/upper bound for the number of keys a node can hold. It it further says that:

1. Every non-root node stores at least t - 1 keys and has t children.
2. Every node stores at most 2*t - 1 keys and has 2*t children.

So you get for t = 2:

1. t - 1 = 1 keys and t = 2 children
2. 2*t - 1 = 3 keys and 4 children

For t = 3

1. t - 1 = 2 keys and t = 3 children
2. 2*t - 1 = 5 keys and 6 children

Now here's the problem: It seems that the nodes in a B-Tree can only store an odd number of keys when they are full.

Why can't there be a node with, let's say at most 4 keys and 5 children? Does it have something to do with splitting the node?

Answer 1

It seems that the nodes in a B-Tree can only store an odd number of keys?

Definitely not. The numbers you have written is minimum and maximum number ofof keys respectively, so for t = 2, nodes with 1, 2, 3 keys are allowed. For t = 3, nodes with 2, 3, 4, 5 keys are allowed.

Moreover, the root of the tree is allowed to have only 1 key.

It is possible to define (and implement) trees that have eg. 1 or 2 keys in a node (so-called 2-3 trees). The reason B-trees are defined to accommodate one more, is that this leads to faster performance. Particularly, this allows amortized O(1) (counting splitting and joining operations) delete and insert operations.