Trees are hierarchical data structures that impose an ordered structure on data. A tree has a root node and may have many child nodes partitioned into subtrees. Binary trees restrict nodes to have at most two children. Binary trees can be represented using arrays or linked lists. Common traversals of binary trees include inorder, preorder, and postorder. Binary search trees organize nodes so that all left descendants are less than the parent and all right descendants are greater. Deletion of nodes in a binary search tree depends on whether the node has zero, one, or two children.

1. TreesData Structures and Algorithms

2. DefinitionTrees are used to impose a hierarchical structure on a collection of data items.A tree is a set of one or more nodes T such that:there is a specially designated node called a rootThe remaining nodes are partitioned into n disjointed set of nodes T1, T2,…,Tn, each of which is a tree.

3. Basic TerminologyDegree of a Node of a Tree The degree of a node of a tree is the number of subtrees having this node as a root. In other words, the degree is the number of descendants of a node. If the degree is zero, it is called a terminal or leaf node of a tree.Degree of a Tree The degree of a tree is defined as the maximum of degree of the nodes of the tree, that is, degree of tree = max (degree(node i) for I = 1 to n)Level of a Node We define the level of the node by taking the level of the root node as 1, and incrementing it by 1 as we move from the root towards the subtrees. So the level of all the descendants of the root nodes will be 2. The level of their descendants will be 3, and so on. We then define the depth of the tree to be the maximum value of the level of the node of the tree.

4. Binary TreeA binary tree is a special case of tree as defined in the preceding section, in which no node of a tree can have a degree of more than 2. Therefore, a binary tree is a set of zero or more nodes T such that:there is a specially designated node called the root of the treethe remaining nodes are partitioned into two disjointed sets, T1 and T2, each of which is a binary tree. T1 is called the left subtree and T2 is called right subtree, or vice-versa

5. Skewed Binary Tree

6. Full binary treeA full binary tree is a binary of depth k having 2k − 1 nodes. If it has < 2k − 1, it is not a full binary tree. For example, for k = 3, the number of nodes = 2k − 1 = 23 − 1 = 8 − 1 = 7. A full binary tree with depth k = 3 is shown

7. Complete binary treeA complete binary tree of depth k is a tree with n nodes in which these n nodes can be numbered sequentially from 1 to n, as if it would have been the first n nodes in a full binary tree of depth k.A complete binary tree with depth k = 3 is shown

8. Representation of Binary TreeBT are represented using 2 waysArray representationLinked list representation

9. 1. Array Representationwe can map node i to the ith index in the array, and the parent of node i will get mapped at an index i/2, whereas the left child of node i gets mapped at an index 2i and the right child gets mapped at an index 2i + 1

10. 2. Linked List representationwe can use a linked representation, in which every node is represented as a structure with three fields: one for holding data, one for linking it with the left subtree, and the third for linking it with right subtree

11. Traversal of Binary TreeThere are 6 ways of traversal of binary treeLDRLRDDLRRDLRLDDRLWhere,L stands for traversing the left subtree, R stands for traversing the right subtree,D stands for processing the data of the node

12. The order LDR is called as inorder;the order LRD is called as postorder;the order DLR is called as preorder. The remaining three orders are not used.

13. Binary Search TreeA binary search tree is a binary tree that may be empty, and every node must contain an identifier. An identifier of any node in the left subtree is less than the identifier of the root. An identifier of any node in the right subtree is greater than the identifier of the root. Both the left subtree and right subtree are binary search trees

14. DELETION OF A NODE FROM BINARY SEARCH TREEDeletion of a node with two childrenDeletion of a Node with One ChildDeletion of a Node with No Child

15. Deletion of a node with two children

16. Deletion of a Node with One Child

17. Deletion of a Node with No Child

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