Let T be a rooted tree with n nodes.

Given a random graph of n nodes and an average degree .

A tree with n nodes has exactly n 1 branches or degree.

However, it is expensive to grow and wastes space proportional to 2 - n for a tree of depth h with n nodes.

A red black tree which contains n internal nodes has a height of O(log(n)).

Denote by the number of binary trees on n nodes.

Finally, the sampled sub-graph is expanded to include all of the edges that exist in the network between these n nodes.

It follows that for a tree with n nodes and height h:

For example, when the items are inserted in sorted key order, the tree degenerates into a linked list with n nodes.

In a fully connected network with n nodes, there are n(n-1)/2 direct links.