The document summarizes 2-3 trees, which are binary search trees where each internal node has either 2 or 3 children. It describes the structure of internal and leaf nodes, and provides algorithms for searching, inserting, and deleting elements in 2-3 trees. Searching follows the structure of the tree to guide the search path. Insertion may split nodes or increase the tree height, while deletion may merge nodes or decrease the height.

1. 5.3 2-3 Trees
Definition
• A 2-3 Tree is a null tree (zero nodes) or a single node tree (only one node) or a
multiple node tree with the following properties:
1.
Each interior node has two or three children
2.
Each path from the root to a leaf has the same length.
Fields of a Node :
Internal Node
p1 k1 p2 k2 p3
p1 : Pointer to the first child
p2 : Pointer to the second child
p3 : Pointer to the third child
k1 : Smallest key that is a descendent of the second child
k2 : Smallest key that is a descendent of the third child
Leaf Node
key other fields
Records are placed at the leaves. Each leaf contains a record (and key)
Example: See Figure 5.23
Figure 5.23: An example of a 2-3 tree

2. Search
• The values recorded at the internal nodes can be used to guide the search path.
• To search for a record with key value x, we first start at the root. Let k1 and k2 be
the two values stored here.
1.
If x < k1, move to the first child
2.
If x k1 and the node has only two children, move to the second child
3.
If x k1 and the node has three children, move to the second child if x < k2 and
to the third child if x k2.
• Eventually, we reach a leaf. x is in the tree iff x is at this leaf.
Path Lengths
• A 2-3 Tree with k levels has between 2k - 1 and 3k - 1 leaves
• Thus a 2-3 tree with n elements (leaves) requires
at least 1 + log3n levels
at most 1 + log2n levels
5.3.1 2-3 Trees: Insertion
For an example, see Figure 5.24.

3. Figure 5.24: Insertion in 2-3 trees: An example
Figure 5.25: Deletion in 2-3 trees: An Example

5. Insert (x)
1.
Locate the node v, which should be the parent of x
2.
If v has two children,
• make x another child of v and place it in the proper order
• adjust k1 and k2 at node v to reflect the new situation
3.
If v has three children,
• split v into two nodes v and v'. Make the two smallest among four children
stay with v and assign the other two as children of v'.
• Recursively insert v' among the children of w where
w = parent of v
• The recursive insertion can proceed all the way up to the root, making it
necessary to split the root. In this case, create a new root, thus increasing
the number of levels by 1.
5.3.2 2-3 Trees: Deletion
Delete (x)
1.
Locate the leaf L containing x and let v be the parent of L
2.
Delete L. This may leave v with only one child. If v is the root, delete v and let its
lone child become the new root. This reduces the number of levels by 1. If v is not
the root, let
p = parent of v
If p has a child with 3 children, transfer an appropriate one to vif this child is
adjacent (sibling) to v.
If children of p adjacent to v have only two children, transfer the lone child of v to
an adjacent sibling of v and delete v.
• If p now has only one child, repeat recursively with p in place of v. The
recursion can ripple all the way up to the root, leading to a decrease in the
number of levels.
Example: See Figure 5.25.