Recall that, for binary-search trees, although the average-case times for the lookup, insert, and delete methods are all O(log N), where N is the number of nodes in the tree, the worst-case time is O(N). We can guarantee O(log N) time for all three methods by using a balanced tree -- a tree that always has height O(log N)-- instead of a binary-search tree.

A number of different balanced trees have been defined, including AVL trees, red-black trees, and B trees. You might learn about the first two in an algorithms class, and the third in a database class. Here we will look at yet another kind of balanced tree called a 2-3 Tree.

The important idea behind all of these trees is that the insert and delete operations may restructure the tree to keep it balanced. So lookup, insert, and delete will always be logarithmic in the number of nodes, but insert and delete may be more complicated than for binary-search trees.

The important facts about a 2-3 tree are:

• Every non-leaf node has either 2 or 3 children.
• All leaves are at the same depth.
• Information (keys and associated data) is stored only at leaves (internal nodes are for organization only).
• Keys at leaves are ordered left to right.
• In addition to child pointers, each internal node stores:
  • the value of the max key in the left subtree (leftMax)
  • the value of the max key in the middle subtree (middleMax)
• If a node only has 2 children, they are left and middle (not left and right) children.

As for binary search trees, the same values can usually be represented by more than one tree. Here are three different 2-3 trees that all store the values 2,4,7,10,12,15,20,30:

                      ------------
                      | 4  |  12 |
                      ------------
                     /     |       \
		    /      |        \
      ------------    ------------    -------------
      | 2  |  4  |    | 7  | 10  |    | 15  | 20  |
      ------------    ------------    -------------
      /    |          /    |     \    /      |     \
     2     4         7    10     12  15      20     30



                      ------------
                      | 7  |  15  |
                      ------------
                    /      |        \
		   /       |         \
      ------------   --------------   ---------------
      | 2  |  4  |   |  10  | 12  |   |  20  |  30  |
      ------------   --------------   ---------------
      /    |     \    /     |    \      |    |
     2     4      7  10     12   15    20    30


                                       ---------------
				       |  10  |  30  |
                                       ---------------
				      /       |
                     -----------------        \
		    /                          \
               ------------                     ---------------
               | 4  |  10 |                     |  15  |  30  |
               ------------                     ---------------
              /      |                          /      |
 	     /       |                         /       |
  ------------   -------------   ---------------  ---------------
  | 2  |  4  |   |  7  | 10  |   |  12  |  15  |  |  20  |  30  |
  ------------   -------------   ---------------  ---------------
  /    |         /    |           /     |          /     |
 2     4        7     10         12    15         20    30

Operations on a 2-3 Tree

1. T is empty: return false
2. T is a leaf node: return true iff the key value in T is k

1. k <= T.leftMax: look up k in T's left subtree
2. T.leftMax < k <= T.middleMax: look up k in T's middle subtree
3. T.middleMax < k: look up k in T's right subtree

           if T is empty replace it with a single node containing k
           else if T is just 1 node m:
	        (a) create a new leaf node n containing k
		(b) create a new internal node with m and n as its children,
		    and with the appropriate values for leftMax and middleMax
           else call auxiliary method insert(T, k)

The auxiliary insert method performs the following steps to find node n, the parent of the new node:

• base case: T's children are leaves - n is found! (T will be the parent of the new node)
• recursive cases:
  • k < T.leftMax: insert k into T's left subtree
  • T.leftMax < k < T.middleMax, or T only has 2 children: insert k into T's middle subtree
  • k > T.middleMax and T has 3 children: insert k into T's right subtree

Case 1: n has only 2 children

• Insert k as the appropriate child of n:
  1. if k < n.leftMax, then make k n's left child (move the others over), and fix the values of n.leftMax and n.middleMax. Note that all ancestors of n still have correct values for their leftMax and middleMax fields (because the new value is not the "max" child of n).
  2. if k is between n.leftMax and n.middleMax, then make k n's middle child and fix the value of n.middleMax. Again, no ancestors of n need to have their fields changed.
  3. if k > n.middleMax, then make k n's right child and fix the leftMax or middleMax fields of n's ancestors as needed.

Case 2: n already has 3 children

• Make k the appropriate new child of n, anyway (fixing the values of n.leftMax and/or n.middleMax as needed). Now n has 4 children.
• Create a new internal node m. Give m n's two rightmost children and set the values of m.leftMax and m.middleMax.
• Add m as the appropriate new child of n's parent (i.e., add m just to the right of n). If n's parent had only 2 children, then stop creating new nodes, just fix the values of the leftMax and middleMax fields of ancestors as needed. Otherwise, keep creating new nodes recursively up the tree. If the root is given 4 children, then create a new node m as above, and create a new root node with n and m as its children.

What is the time for insert? Finding node n (the parent of the new node) involves following a path from the root to a parent of leaves. That path is O(height of tree) = O(log N), where N is the number of nodes in the tree (recall that it is also log M, where M is the number of key values stored in the tree).

Once node n is found, finishing the insert, in the worst case, involves adding new nodes and/or fixing fields all the way back up from the leaf to the root, which is also O(log N).

So the total time is O(log N), which is also O(log M).

Once node n (the parent of the node to be deleted) is found, there are two cases, depending on how many children n has:

case 1: n has 3 children

• Remove the child with value k, then fix n.leftMax, n.middleMax, and n's ancestors' leftMax and middleMax fields if necessary.

case 2: n has only 2 children

• If n is the root of the tree, then remove the node containing k. Replace the root node with the other child (so the final tree is just a single leaf node).
• If n has a left or right sibling with 3 kids, then:
  • remove the node containing k
  • "steal" one of the sibling's children
  • fix n.leftMax, n.middleMax, and the leftMax and middleMax fields of n's sibling and ancestors as needed.
• If n's sibling(s) have only 2 children, then:
  • remove the node containing k
  • make n's remaining child a child of n's sibling
  • fix leftMax and middleMax fields of n's sibling as needed
  • remove n as a child of its parent, using essentially the same two cases (depending on how many children n's parent has) as those just discussed

The time for delete is similar to insert; the worst case involves one traversal down the tree to find n, and another "traversal" up the tree, fixing leftMax and middleMax fields along the way (the traversal up is really actions that happen after the recursive call to delete has finished).

So the total time is 2 * height-of-tree = O(log N).

• keys are stored only at leaves, ordered left-to-right
• non-leaf nodes have 2 or 3 children (never 1)
• non-leaf nodes also have leftMax and middleMax values (as well as pointers to children)
• all leaves are at the same depth
• the height of the tree is O(log N), where N = # nodes in tree
• at least half the nodes are leaves, so the height of the tree is also O(log M) for M = # values stored in tree
• the lookup, insert, and delete methods can all be implemented to run in time O(log N), which is also O(log M)