1 Basic Operation Of Tree Data Structure

• Create – create a tree in the data structure.
• Insert − Inserts data in a tree.
• Search − Searches specific data in a tree to check whether it is present or not.
• Traversal:
  • Preorder Traversal – perform Traveling a tree in a pre-order manner in the data structure.
  • In order Traversal – perform Traveling a tree in an in-order manner.
  • Post-order Traversal –perform Traveling a tree in a post-order manner.

2 Properties of Tree Data Structure

Consider the following binary tree:

        1
       / \
      2   3
     / \
    4   5

1 Number of Edges:

An edge connects two nodes in a tree. If a tree has N nodes, it will have (N-1) edges. There is exactly one path from each node to any other node in the tree.

• The tree has 5 nodes, so it will have (5 - 1) = 4 edges.

2 Depth of a Node:

The depth of a node is the length of the path from the root to that node. Each edge along the path adds 1 unit to the depth. Alternatively, it's the number of edges in the path from the root to the node.

• Depth of node 1: 0 (root node)
• Depth of node 2: 1
• Depth of node 3: 1
• Depth of node 4: 2
• Depth of node 5: 2

3 Height of a Node:

The height of a node is the length of the longest path from the node to a leaf node of the tree. It represents the longest downward path from the node to a leaf.

• Height of node 1: 2 (longest path to leaf: 1 → 2 → 4)
• Height of node 2: 1 (longest path to leaf: 2 → 4)
• Height of node 3: 1 (longest path to leaf: 3)
• Height of node 4: 0 (leaf node)
• Height of node 5: 0 (leaf node)

4 Height of the Tree:

The height of a tree is the length of the longest path from the root to a leaf node of the tree. It represents the longest downward path in the entire tree.

• The longest path from the root to a leaf node is 2 (root → 2 → 4 or root → 2 → 5), so the height of the tree is 2.

5 Degree of a Node:

The degree of a node is the total number of subtrees attached to that node. For a leaf node, the degree is 0. The degree of a tree is the maximum degree among all the nodes in the tree.

• Degree of node 1: 2 (two subtrees: 2 and 3)
• Degree of node 2: 2 (two subtrees: 4 and 5)
• Degree of node 3: 0 (leaf node)
• Degree of node 4: 0 (leaf node)
• Degree of node 5: 0 (leaf node)

3 Need for Tree Data Structure

1 Hierarchical Representation:

Trees are well-suited for representing hierarchical relationships, such as organizational structures, file systems, XML/HTML documents, and nested data.

2 Efficient Searching and Sorting:

Binary search trees (BSTs) provide efficient searching, insertion, and deletion operations with an average time complexity of O(log n), making them ideal for implementing search and sort algorithms.

3 Data Organization:

Trees can organize data in a structured and efficient manner, enabling quick access and manipulation of data elements.

4 Binary Heap:

Binary heaps, a type of binary tree, are used to implement priority queues, which are essential in algorithms like Dijkstra's shortest path algorithm and heap sort.

5 Representation of Arithmetic Expressions:

Expression trees are used to represent arithmetic expressions in a tree format, facilitating evaluation and manipulation of expressions.

6 Efficient Storage and Retrieval:

Tries (prefix trees) are specialized tree structures used for efficient storage and retrieval of strings, making them suitable for applications like autocomplete and spell checking.

7 Balancing and Optimization:

Self-balancing trees like AVL trees and Red-Black trees maintain balance in the tree structure, ensuring efficient performance of operations like insertion, deletion, and search.

8 Graph Representation:

Trees are a special type of graph with no cycles, making them suitable for representing hierarchical relationships and providing a foundation for graph algorithms and data structures.

9 Algorithm Design:

Many algorithms and data structures, such as depth-first search (DFS) and breadth-first search (BFS), are based on tree traversal techniques and concepts.

10 Memory Management:

Trees are used in memory management algorithms like binary buddy systems and binary trees for memory allocation and deallocation.

4 Relationship between number of nodes (n) and the height (h) in binary tree

#4.1 Maximum Number of Nodes at a Given Height:

• In a binary tree, the height h represents the longest path from the root node to any leaf node.
• At each level of the tree, the number of nodes that can exist doubles.
• At level 0 (the root level), there is 1 node.
• At level 1, there can be at most 2 nodes.
• At level 2, there can be at most 4 nodes.
• And so on…

Therefore, the maximum number of nodes at a given height is 2^h.

#4.2 Maximum Height with a Given Number of Nodes:

• Conversely, if we know the number of nodes in a binary tree, we can determine the maximum height of the tree.
• For example, if there is only 1 node, the tree's height is 0 (h = 0).
• With 2 nodes, the tree's height is 1 (h=1).
• With 3 to 4 nodes, the tree's height is 2 (h = 2).
• With 5 to 8 nodes, the tree's height is 3 (h = 3).
• In general, if there are n  nodes, the height of the tree is log2(n), rounded up to the nearest integer.

#4.3 Relationship between Number of Nodes and Height:

We can express the relationship between the number of nodes (n) and the height (h) of a binary tree with the equation:

n = 2^(h+1)-1

Or equivalently:

h = log2(n+1) -1

#4.4 Maximum and minimum height

Maximum Height:

• The maximum height of a binary tree occurs when the tree is completely unbalanced, with each node having only one child, forming a linked list structure.
• In such a scenario, the height of the tree is equal to the number of nodes minus one.
• Therefore, the maximum height of a binary tree with n nodes is h = n - 1.

Minimum Height:

• The minimum height of a binary tree occurs when the tree is perfectly balanced, with each internal node having two children, and the height is minimized.
• For a binary tree with n nodes, the minimum height is achieved in a balanced binary tree, where the number of nodes in the left and right subtrees of any node differs by at most one.
• In a perfectly balanced binary tree, the height is approximately log₂(n), rounded up to the nearest integer.
• Therefore, the minimum height of a binary tree with n nodes is h = ⌈log₂(n + 1)⌉ - 1, where ⌈x⌉ represents the ceiling function, rounding x up to the nearest integer.

Example:

Suppose we have a binary tree with 15 nodes:

• Maximum Height: h = 15 - 1 = 14
• Minimum Height (for a perfectly balanced tree): h = ⌈log₂(15 + 1)⌉ - 1 = ⌈log₂(16)⌉ - 1 = ⌈4⌉ - 1 = 4 - 1 = 3