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# 23 Tree Best Part

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### 23 Tree Best Part

1. 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. 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. 3. Figure 5.24: Insertion in 2-3 trees: An example Figure 5.25: Deletion in 2-3 trees: An Example
4. 4. 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.