Tree Data Structure-
Tree data structure may be defined as-
| Tree is a non-linear data structure which organizes data in a hierarchical structure and this is a recursive definition. OR A tree is a connected graph without any circuits. OR If in a graph, there is one and only one path between every pair of vertices, then graph is called as a tree. |
Example-
Properties-
The important properties of tree data structure are-
- There is one and only one path between every pair of vertices in a tree.
- A tree with n vertices has exactly (n-1) edges.
- A graph is a tree if and only if it is minimally connected.
- Any connected graph with n vertices and (n-1) edges is a tree.
Tree Terminology-
The important terms related to tree data structure are-
1. Root–
- The first node from where the tree originates is called as a root node.
- In any tree, there must be only one root node.
- We can never have multiple root nodes in a tree data structure.
Example-
Here, node A is the only root node.
2. Edge–
- The connecting link between any two nodes is called as an edge.
- In a tree with n number of nodes, there are exactly (n-1) number of edges.
Example-
3. Parent–
- The node which has a branch from it to any other node is called as a parent node.
- In other words, the node which has one or more children is called as a parent node.
- In a tree, a parent node can have any number of child nodes.
Example-
Here,
- Node A is the parent of nodes B and C
- Node B is the parent of nodes D, E and F
- Node C is the parent of nodes G and H
- Node E is the parent of nodes I and J
- Node G is the parent of node K
4. Child–
- The node which is a descendant of some node is called as a child node.
- All the nodes except root node are child nodes.
Example-
Here,
- Nodes B and C are the children of node A
- Nodes D, E and F are the children of node B
- Nodes G and H are the children of node C
- Nodes I and J are the children of node E
- Node K is the child of node G
5. Siblings–
- Nodes which belong to the same parent are called as siblings.
- In other words, nodes with the same parent are sibling nodes.
Example-
Here,
- Nodes B and C are siblings
- Nodes D, E and F are siblings
- Nodes G and H are siblings
- Nodes I and J are siblings
6. Degree–
- Degree of a node is the total number of children of that node.
- Degree of a tree is the highest degree of a node among all the nodes in the tree.
Example-
Here,
- Degree of node A = 2
- Degree of node B = 3
- Degree of node C = 2
- Degree of node D = 0
- Degree of node E = 2
- Degree of node F = 0
- Degree of node G = 1
- Degree of node H = 0
- Degree of node I = 0
- Degree of node J = 0
- Degree of node K = 0
7. Internal Node–
- The node which has at least one child is called as an internal node.
- Internal nodes are also called as non-terminal nodes.
- Every non-leaf node is an internal node.
Example-
Here, nodes A, B, C, E and G are internal nodes.
8. Leaf Node–
- The node which does not have any child is called as a leaf node.
- Leaf nodes are also called as external nodes or terminal nodes.
Example-
Here, nodes D, I, J, F, K and H are leaf nodes.
9. Level-
- In a tree, each step from top to bottom is called as level of a tree.
- The level count starts with 0 and increments by 1 at each level or step.
Example-
10. Height-
- Total number of edges that lies on the longest path from any leaf node to a particular node is called as height of that node.
- Height of a tree is the height of root node.
- Height of all leaf nodes = 0
Example-
Here,
- Height of node A = 3
- Height of node B = 2
- Height of node C = 2
- Height of node D = 0
- Height of node E = 1
- Height of node F = 0
- Height of node G = 1
- Height of node H = 0
- Height of node I = 0
- Height of node J = 0
- Height of node K = 0
11. Depth-
- Total number of edges from root node to a particular node is called as depth of that node.
- Depth of a tree is the total number of edges from root node to a leaf node in the longest path.
- Depth of the root node = 0
- The terms “level” and “depth” are used interchangeably.
Example-
Here,
- Depth of node A = 0
- Depth of node B = 1
- Depth of node C = 1
- Depth of node D = 2
- Depth of node E = 2
- Depth of node F = 2
- Depth of node G = 2
- Depth of node H = 2
- Depth of node I = 3
- Depth of node J = 3
- Depth of node K = 3
12. Subtree-
- In a tree, each child from a node forms a subtree recursively.
- Every child node forms a subtree on its parent node.
Example-
13. Forest-
A forest is a set of disjoint trees.
Example-
Binary Tree-
| Binary tree is a special tree data structure in which each node can have at most 2 children.Thus, in a binary tree,Each node has either 0 child or 1 child or 2 children. |
Example-
Unlabeled Binary Tree-
| A binary tree is unlabeled if its nodes are not assigned any label. |
Example-
Consider we want to draw all the binary trees possible with 3 unlabeled nodes.
Using the above formula, we have-
Number of binary trees possible with 3 unlabeled nodes
= 2 x 3C3 / (3 + 1)
= 6C3 / 4
= 5
Thus,
- With 3 unlabeled nodes, 5 unlabeled binary trees are possible.
- These unlabeled binary trees are as follows-
Labeled Binary Tree-
| A binary tree is labeled if all its nodes are assigned a label. |
Example-
Consider we want to draw all the binary trees possible with 3 labeled nodes.
Using the above formula, we have-
Number of binary trees possible with 3 labeled nodes
= { 2 x 3C3 / (3 + 1) } x 3!
= { 6C3 / 4 } x 6
= 5 x 6
= 30
Thus,
- With 3 labeled nodes, 30 labeled binary trees are possible.
- Each unlabeled structure gives rise to 3! = 6 different labeled structures.
Similarly,
- Every other unlabeled structure gives rise to 6 different labeled structures.
- Thus, in total 30 different labeled binary trees are possible.
Types of Binary Trees-
Binary trees can be of the following types-
- Rooted Binary Tree
- Full / Strictly Binary Tree
- Complete / Perfect Binary Tree
- Almost Complete Binary Tree
- Skewed Binary Tree
1. Rooted Binary Tree-
A rooted binary tree is a binary tree that satisfies the following 2 properties-
- It has a root node.
- Each node has at most 2 children.
Example-
2. Full / Strictly Binary Tree-
- A binary tree in which every node has either 0 or 2 children is called as a Full binary tree.
- Full binary tree is also called as Strictly binary tree.
Example-
Here,
- First binary tree is not a full binary tree.
- This is because node C has only 1 child.
3. Complete / Perfect Binary Tree-
A complete binary tree is a binary tree that satisfies the following 2 properties-
- Every internal node has exactly 2 children.
- All the leaf nodes are at the same level.
Complete binary tree is also called as Perfect binary tree.
Example-
Here,
- First binary tree is not a complete binary tree.
- This is because all the leaf nodes are not at the same level.
4. Almost Complete Binary Tree-
An almost complete binary tree is a binary tree that satisfies the following 2 properties-
- All the levels are completely filled except possibly the last level.
- The last level must be strictly filled from left to right.
Example-
Here,
- First binary tree is not an almost complete binary tree.
- This is because the last level is not filled from left to right.
5. Skewed Binary Tree-
A skewed binary tree is a binary tree that satisfies the following 2 properties-
- All the nodes except one node has one and only one child.
- The remaining node has no child.
OR
A skewed binary tree is a binary tree of n nodes such that its depth is (n-1).
Types of Skewed Binary trees
There are 2 special types of skewed tree:
- Left Skewed Binary Tree:
These are those skewed binary trees in which all the nodes are having a left child or no child at all. It is a left side dominated tree. All the right children remain as null.
- Right Skewed Binary Tree:
These are those skewed binary trees in which all the nodes are having a right child or no child at all. It is a right side dominated tree. All the left children remain as null.
Example-
