How does the binary search tree algorithm work?

The binary search tree algorithm works by placing nodes with lesser values to the left and greater values to the right.

A binary search tree (BST) is a tree data structure in which each node has at most two children, referred to as the left child and the right child. For each node, all elements in the left subtree are less than the node, and all elements in the right subtree are greater than the node. This property makes the BST an ordered or sorted binary tree.

The algorithm for inserting a node into a BST is quite straightforward. Starting from the root, the algorithm compares the new node's value with the current node's value. If the new value is less than the current node's value, the algorithm moves to the left child; if it's greater, it moves to the right child. This process is repeated until the algorithm finds an empty spot where the new node can be inserted.

Searching for a value in a BST follows a similar process. Starting from the root, the algorithm compares the search value with the current node's value. If the search value is less than the current node's value, the algorithm moves to the left child; if it's greater, it moves to the right child. If the search value is equal to the current node's value, the search is successful. If the algorithm reaches a null node (i.e., the algorithm has moved to a child of a leaf node), the search is unsuccessful.

Deleting a node from a BST is a bit more complex because it requires maintaining the BST property. If the node to be deleted has no children, it can simply be removed. If the node has one child, it can be replaced with its subtree. If the node has two children, it can be replaced with either its in-order predecessor or its in-order successor.

The BST algorithm is efficient for many dynamic set operations, including search, minimum, maximum, predecessor, successor, insert, and delete. The time complexity for these operations is proportional to the height of the tree. For a balanced BST, the height is logarithmic in the number of nodes, so these operations can be performed efficiently. However, for an unbalanced BST, the height can be linear in the number of nodes, in which case these operations can be inefficient.