B-Trees Study Guide
Author: Brandon Lee


Depth We define the depth of a node as how far it is from the root. For consistency, we say the root has a depth of 0.

Height We define the height of a tree as the depth of the deepest node.

Notice that depending on how we insert into our BST, our height could vary drastically. We say a tree is “spindly” if it has height close to N and a tree is “bushy” if its height is closer to logN. For operations such as getting a node, we want to have the height to be as small as possible, thus favoring “bushy” BSTs


Two specific B-Trees in this course are 2-3 Trees (A B-Tree where each node has 2 or 3 children), and 2-3-4/2-4 Trees (A B-Tree where each node has 2, 3, or 4 children). The key idea of a B-Tree is to over stuff the nodes at the bottom to prevent increaseing the height of the tree. This allows us to ensure a max height of logN.

Make sure you know how to insert into a B-Tree. Refer back to lecture slides for examples.

With our restriction on height, we get that the runtime for contains and add are both THETA(LogN)

B-Tree invariants

Because of how we add to our tree, we get two nice invariants for B-Trees:

  1. All leaves must be the same distance from the source
  2. A non-leaf node with k items mut has exactly k+1 children.

Practice Problems

  1. Draw the 2-3 tree that results when you insert the keys A B C D E F G in order.
  2. How many compares does it take in the worst case to decide whether to go left, middle, or right from a 3 node?
  3. Problem 5 of the Fall 2014 midterm.
  4. Problem 1c, e of the Spring 2018 Midterm 2
  5. Problem 8b of the Spring 2016 Midterm 2