Big O for Trees

If a tree is uniform and balanced, we can describe it in terms of several parameters:

• b, the breadth or branching factor, is the number of branches per interior node. For a binary tree, b = 2.

• d is the depth, the height of the tree. d = log_b(n)

• n is the number of leaf nodes. n = b^d

Note that most of the nodes are on the bottom row of the tree. If b = 2, half the nodes are on the bottom; if b is higher, an even greater proportion will be on the bottom.

In general, a tree algorithm will have Big O:

• if one branch of the tree is followed and the others are abandoned, O(log(n)): proportional to depth.

• if all branches of the tree are processed, O(n * log(n))