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123 lines
4.0 KiB
Markdown
123 lines
4.0 KiB
Markdown
---
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title: Trees
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---
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# Trees
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A tree data structure can be defined recursively (locally) as a collection of nodes (starting at a root node), where each node is a data structure consisting of a value, together with a list of references to nodes (the "children"), with the constraints that no reference is duplicated, and none points to the root. A tree with no nodes is called a null or empty tree.
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A binary tree is a non linear data structure consisting nodes, where each node has the following 3 components:
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**Data element**: Stores any kind of data in the node
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**Left pointer**: Points to the sub-tree on the left side of node
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**Right pointer**: Points to the sub-tree on the right side of the node
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As the name suggests, the data element stores any kind of data in the node.
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The left and right pointers point to binary trees on the left and right side of the node respectively.
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If a tree is empty, it is represented by a null pointer.
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## Terminology used in trees:
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**Root** :
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The top node in a tree.
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**Child**:
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A node directly connected to another node when moving away from the Root.
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**Parent**:
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The converse notion of a child.
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**Siblings**:
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A group of nodes with the same parent.
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**Descendant**:
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A node reachable by repeated proceeding from parent to child.
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**Ancestor**:
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A node reachable by repeated proceeding from child to parent.
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**Branch**(internal node):
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A node of a tree that has child nodes.
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**Leaf**(less commonly called External node):
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A node with no children.
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**Degree**:
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The number of subtrees of a node.
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**Edge**:
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The connection between one node and another.
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**Path**:
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A sequence of nodes and edges connecting a node with a descendant.
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**Level**:
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The level of a node is defined by 1 + (the number of connections between the node and the root).
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**Height of tree**:
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The height of a tree is the height of its root node.
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**Depth**:
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The depth of a node is the number of edges from the tree's root node to the node.
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**Forest**:
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A forest is a set of n ≥ 0 disjoint trees.
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### Some Popular Types of Trees:
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- **Binary Tree**. It has the following properties:
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- The maximum number of nodes at level ‘l’ of a binary tree is 2^l-1.
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- In Binary tree where every node has 0 or 2 children, number of leaf nodes is always one more than nodes with two children.
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- **Binary Search Tree**. It has the following properties:
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- The left subtree of a node contains only nodes with keys lesser than the node’s key
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- The right subtree of a node contains only nodes with keys greater than the node’s key
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- The left and right subtree each must also be a binary search tree
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- **AVL Tree**. It has the following properties:
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- Height difference of left and right subtree of node should be less than 2
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- Re-balancing is done when heights of two child subtrees of a node differ by more than one
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* Red Black Tree
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* Splay Tree
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* Huffmann Tree
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* M-Way Search Tree (used for huge amounts of data)
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### Common uses
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* Representing hierarchical data
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* Storing data in a way that makes it easily searchable
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* Representing sorted lists of data
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* Routing algorithms
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### Code of a tree node
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```cpp
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struct node
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{
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int data; //Data element
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struct node * left; //Pointer to left node
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struct node * right; //Pointer to right node
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};
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```
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### Code for node creation
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createNode() returns a new node with the given data and NULL left and right pointers.
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```cpp
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struct node* newNode(int element)
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{
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struct node* temp = (node*)malloc(sizeof(node)); //Allocate memeory for temp node
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temp->data = element; // Assign element to temp
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temp->left = NULL; // Initialize left child as NULL
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temp->right = NULL; // Initialize right child as NULL
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return temp;
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}
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```
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#### More Information:
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* [CMU lesson notes](http://www.cs.cmu.edu/~clo/www/CMU/DataStructures/Lessons/lesson4_1.htm)
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* [Wikipedia](https://en.wikipedia.org/wiki/Tree_(data_structure))
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