Tree – Introduction, Terminologies and Explanation
|A tree is a hierarchical data structure, unlike array and linked list(which are known as a linear data structure).
Saying hierarchy means nodes(or vertices) are ordered in top to down order, i.e. topmost node is referred as Root node.
For example in image below-
- 8 is the root node.
- Node 5 and 10 are said to be children of node 8( immediate parent).
- Node 5 and 10 are the siblings to each other.
Trees and Graphs – Trees can also be generalized as a graph which is acyclic and connected provided any two vertices are connected only through one path.
A tree, with each node having at most 2 children is a Binary Tree.
A tree, with each node having at most 3 children is a Ternary Tree.
A tree, with each node having at most n children is an n-ary Tree.
Note:-
Although there can be an n-ary tree, but in practice, binary trees are more widely used. Further, various modifications of binary trees are also available such as Binary search tree, AVL Tree, Red-Black tree etc.
Table of Contents
Advantages of using Tree:
- Widely used operation searching is better optimized with tree data structure. Like balanced binary search tree always takes O(logn) for searching.
- Insertion to a tree is faster than Linked list and slower than the arrays. Interestingly, a binary tree is also implemented using array referred as Heap.
- In case of deletion, it is like a linked list. Deletion is faster against an array.
Necessary conditions for a tree:
- It should not contain any cycle.
- A node should not contain self-loop.
- A child will have only one parent.
- Any two nodes in a tree are connected through only one possible path.
Terminologies:
- Root node – The topmost node of a tree,
- Leaf node – Node having no children, <B, I, K, F, G, H>
- Internal/parent node – all node except leaf node <C, D, E, J>
- Degree of node X– the number of nodes in neighbor of X or number of subtrees of node X
e.g. degree(E) = 3 {C, I, J}; deg(B) = 1; deg(J) = 2 - Edge – Connectivity between two node
- Path – Sequence of nodes connected one after other through edges.
e.g. A-D-G, A-C-E-I - Height of a node – The longest path from the node X to the reachable leaf node.
e.g. height(C) = 3 {C-E-J-K} - Height of tree – Height of root node is referred as the height of a tree i.e. the maximum no of edges in the greatest path from root node to the leaf node.
e.g. height of the given tree is 4 - Level of tree – Level of a tree is the number of nodes from the node to root.
e.g. level of root A is 1,
level of node B, C, D is 2,
level of node E, F, G, H is 3 - Descendant and Ancestor – As name justified, Descendant of a node is its children and grand children. Similarly, ancestor of a node is its parent and grand parents.
Implementation:
Almost every languages already have the in-built library for trees and it’s multiple implementations as it has very vast applications and is used almost everywhere. In C, it’s implemented using structure. Following is the declaration of a node of a binary tree, left and right pointer represent left subtree and right subtree respectively whereas the data holds the value of the current node.
struct binaryTree{ int data; struct binaryTree *left; // pointing left subtree struct binaryTree *right; // pointing right subtree };
Applications of Trees :-
- Router algorithm
- Social networking based application
- File system, Directory implementation
- Maps typically uses tree data structure
- In games especially, when the requirement is multi-stage decision-making.
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