B-trees

If a tree could speak...

A B-tree is a data structure used for storing data in secondary memory, such as on a hard disk. It has the following properties:

1. Each non-leaf node has at least $t$ children and at most $2t$ children, where $t$ is a parameter known as the branching factor.

2. Each leaf node contains a number of keys, from t-1 to 2t-1.

3. The length of the path from the root to any leaf is the same for all leaves.

Note that the root node can also be a leaf.

Two B-trees are considered different if they differ as graphs with labeled vertices, or if a vertex with the same labels contains a different number of keys. For instance, there are $8$ distinct B-trees with $4$ keys and a branching factor of $2$:

Determine the number of distinct B-trees that can be formed with $n$ keys in the leaves and a branching factor of $t$.