BST (Binary Search Tree) #definition

• Root - starting Node

• Each Node has a left, and a right "child" - another Node

  • Left child is always "smaller" than the parent
  • Right child is always "greater" than the parent

• Node without children is called a Leaf

• Repetitions are usually not allowed
  This structure forms an ever widening tree-like pattern

• Branch is a "path" from the root to one of the leaves

• Tree is called balanced if length of all it's branches is the same, in this case, length of branches is called the Height of the tree

Operations on BST

Search

This operation is usually recursive, stepping to the left or to the right each call until we find the Node or get to a leaf

public Node searchBST(Node root, Node toFind) {
	if (root == null) {
		return root;
	}
	if (toFind == root) {
		return root;
	} else if (toFind < root) {
		return this.searchBST(root.left, toFind);
	} else {
		return this.searchBST(root.right, toFind);
	}
}

Insertion

This operation is usually recursive, stepping to the left or to the right each call until we get to the leaf

public Node insertNodeBST(Node root, Node new) {
	if (root == null) {
		// new becomes root
		return new;
	}
	if (new < root) {
		root.left = this.insertNodeBST(root.left, new);
	} else if (new > root) {
		root.left = this.insertNodeBST(root.right, new);
	}
	return root;
}

Deletion

This operation is usually recursive, searching for an element to delete and, when found, adjusting tree as follows:

• If the Node to delete has one child - it is just replaced by it
• If the Node has two children - it's immediate successor replaces it

public Node removeNodeBST(Node root, Node toDelete)
	if (!root) {
        return root;
    }
    if (toDelete < root) {
        root.left = removeNodeBST(root.left, toDelete);
    } else if (toDelete > root) {
        root.right = removeNodeBST(root.right, toDelete);
    } else {
        // one right child or zero children
        if (root.left == null) {
            return root.right;
        }
        // one left child or zero children
        if (root.right == null) {
	        return root.left;
        }
        // two children
        Node successor = root.right;
        while (successor.left) {
            successor = successor.left;
        }
        // copy successor data to root
        root.data = successor.data;
        // delete successor
        root.right = removeNodeBST(root.right, successor);
    }
    return root;

BST Traversals

• InOrder Traversal (Left, Root, Right) - visits nodes in ascending order, sorted so to say
• PreOrder Traversal (Root, Left, Right) - useful for creating a copy of the tree
• PostOrder Traversal (Left, Right, Root) - used for deleting the tree efficiently
  BST traversals are usually recursive but can be implemented using a stack instead

Concatenation of BSTs

Concatenation is the process of merging two BSTs, T1​ and T2​, into a single BST while preserving the BST property.
Conditions:

• Given two BSTs:
  • T1​ contains elements smaller than all elements in T2​.
  • T2​ contains elements greater than all elements in T1​.
• An optional pivot element may be provided to maintain balance.

Algorithm for Concatenation

Case 1: The height of the trees is approximately equal

1. Find the maximum element in T1​.
2. Remove this maximum element from T1​.
3. Make the removed element the new root.
4. Attach T1 as the left subtree and T2 as the right subtree.

Case 2: One tree is significantly taller than the other

1. Start at the root of the taller tree (let’s say T1​).
2. Traverse down the right spine until reaching a node where the height of the subtree matches that of T2​.
3. Attach T2​ as the right child of this node.
4. Rebalance the tree if necessary.

Splitting the BST

Splitting a BST means dividing it into two separate BSTs based on a given split value x. The goal is:

• T1​ should contain all elements ≤ x.
• T2​ should contain all elements > x.

Algorithm for Splitting a BST at Value x

1. Search for x in the BST:
   • If x exists, remove it and make it a pivot.
   • If x doesn’t exist, proceed with the nearest node.
2. Recursive Splitting:
   • If x is smaller than the root:
     • Recursively split the left subtree.
     • Attach the right subtree of the split to the right subtree of the root.
   • If x is larger than the root:
     • Recursively split the right subtree.
     • Attach the left subtree of the split to the left subtree of the root.
3. Balancing Step (if needed):
   • If an AVL Tree or Red-Black Tree is used, rebalancing may be required.