B+ Tree order of 1 & 2

Question

I am confused regarding the maximum and minimum number of keys to be inserted in a B+ Tree having an order of 1 and order of 2.

In the videos I watch, it is said that the maximum number of keys to be inserted in a node (except the root) is at least m and at most 2m (assuming m is the order).

According to these 2 statements, what is the minimum & maximum number of keys to be inserted in a B+ Tree, having an order of 1 and order of 2? I am not sure if the 2 statements above conflict, or I misunderstood something. Any idea?

Answer 1

Without having the reference to the video, it looks like they use a non-standard definition of the term order , which is the cause of the confusion.

The standard definition of order for a tree would be the maximum branching factor, ie the maximum number of children that a node may have. So, in that definition it is not the minimum , but the maximum , and it is not about the number of keys , but about the number of children .

The video's definition would mean that the maximum number of keys would always be an even number, while in reality there is no such requirement. B+ trees may well have a maximum branching factor that is even, making the maximum number of keys odd.

Using standard definition of the term order , we specifically have for B+ trees these constraints:

• Its internal nodes have at the most m children. This means that they have at the most m - 1 keys.
• Its internal nodes have at least m/2 (rounded upwards) children, except for the root:
• If its root is not a leaf, it may have as few as 2 children.
• Its leaves contain the actual data values.

Here is an example B+ tree with order 4 (standard definition), which corresponds to a B+ tree where the number of keys must be between 1 and 3 -- something that does not fit with the video's definition:

As you can see, a node can here at most have 4 children, and at most 3 keys. In your definition where 2m represents the maximum number of keys, the order is actually 2m+1 . So you are asking for examples of B+ trees of order 3 and 5, using the standard definition of order .

Here is an example of order 3 -- the lowest possible order for B+ trees -- which means the number of keys must be either 1 or 2 in each node: