In the previous lesson, we considered a particular kind of a binary tree called a Binary Search Tree BST. A binary tree is a binary search tree BST if and only if an inorder traversal of the binary tree results in a sorted sequence. The idea of a binary search tree is that data is stored according to an order, so that it can be retrieved very efficiently. Draw the binary tree which would be created by inserting the following numbers in the order given.

Consider an arbitrary BST of the height k. The total possible number of nodes is given by. Now, assume that we know the number of nodes and we want to figure out the number of comparisons. We have to solve the following equation with respect to k:. This is the most important fact you need to know about BSTs. But building a BST as a balanced tree is not a trivial task.

Binary search trees work well for many applications one of them is a dictionary or help. But they can be limiting because of their bad worst-case performance.

Imagine a binary search tree created from a list that is already sorted. Clearly, the tree will grow to the right or to the left. A binary search tree with this worst. A great care needs to be. There are many techniques for. We will discuss AVL trees in the. Splay Trees will be discussed in advanced data structure courses like A BST is a binary tree of nodes ordered in the following way: Here is an example of a BST Exercise: We have to solve the following equation with respect to k: A binary search tree with this worst- case structure is no more efficient than a regular linked list.

A great care needs to be taken in order to keep the tree as balanced as possible. There are many techniques for balancing a tree including AVL trees, and Splay Alforex woodland. We will discuss AVL trees in the next lesson. We continue this process each node is a root for stock market exponential growth sub tree until we find a null pointer or leaf node where we cannot go any further.

We then insert the node as a left or right child of the leaf node based on node is less or greater than the leaf node. We note that a new node is always inserted as a leaf node.

Binary search tree - Wikipedia

A recursive algorithm for inserting a node into a BST is as follows. Assume we insert a node N to tree T. Otherwise, the make money with your isp delete operation binary search tree inserting is reduced to inserting the node N to left of right sub trees of T, depending on N is less or greater than T. A definition is as follows.

We start from root, and then go left or right until we find or not find the node. A recursive definition of search is as follows.

If the node is equal to root, then we return true. If the root is null, then we return false. Otherwise we recursively solve the problem for T. A recursive definition is as follows. Search should return a true or false, depending on the node is found or not. For example, each node has a parent, unless node is the root.

The best way to deal with deletion seems to be considering special cases. What if the node to delete is a leaf node? What if the node is a node with just one child?

What if the node is an internal node with two children. The latter case is the hardest to resolve.

But we will find a way to handle this situation as well. The node to delete is a leaf node This is a very easy case. Just delete the node.

The node to delete is a node with one child. This is also not too bad. If the node to be deleted is a left child of the parent, then we connect the left pointer of the parent of the deleted node to the single child.

Otherwise if the node to be deleted is a right child of the parent, then we connect the right pointer of the parent of the deleted node to single child. The node to delete is a node with two children This is a difficult case as we need to deal with two sub trees. But we find an easy way to handle it. First we find a replacement node from leaf node or nodes with one child for the node to be deleted.

algorithm - Deletion procedure for a Binary Search Tree - Stack Overflow

We need to do this while maintaining the BST order property. We can easily find this as follows. If the node to be deleted is N, the find the largest node in the left sub tree of N or the smallest node in the right sub tree of N. These are two candidates that can replace the node to be deleted without losing the order property. For example, consider the following tree and suppose we need to delete the root Then we find the largest node in the left sub tree 15 or smallest node in the right sub tree 45 and replace the root with that node and then delete that node.

The following set of images demonstrates this process.