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Types of Binary Trees


Types of Binary Trees

Welcome back to our tutorial series on Data Structures and Algorithms! So far, we've laid the groundwork by understanding the basics of Trees and Binary Trees and then delved into some key Properties of Binary Trees.

We know that a binary tree is a tree where each node has at most two children. But even with this constraint, binary trees can have vastly different structures, impacting how we use them and how efficient operations on them are. Today, we'll classify binary trees into several types based on their structural properties.

Let's look at the common types of binary trees:

1. Full Binary Tree

A Full Binary Tree (sometimes also called a Strictly Binary Tree) is a binary tree where every node has either 0 or 2 children. No node in a full binary tree has only one child.

Intuition:

This structure implies that branches don't just "stop" with a single child; they either continue with two children or terminate completely at a leaf. This creates a somewhat "bushy" appearance where every level is either fully expanded or ends cleanly with leaves.

Full Binary Tree Diagram

2. Complete Binary Tree

A Complete Binary Tree is a binary tree where:

  • All levels are completely filled, except possibly the last level.
  • If the last level is not completely filled, all nodes in the last level are as left as possible.

Intuition:

Complete binary trees are important because they are the most "space-efficient" in terms of node placement for a given height. They are packed as tightly as possible from the top down and from left to right. This structure is particularly useful for implementing data structures like Heaps.

Example:

Imagine filling a binary tree level by level, from left to right, like reading a book. A complete binary tree fills up all spots on a level before moving to the next, and on the last level, it fills from the left first.

Complete Binary Tree Example

Complete Binary Tree Example

Non Complete Binary Tree Example

Non-Complete Binary Tree Example

3. Perfect Binary Tree

A Perfect Binary Tree is a binary tree where all internal nodes have exactly 2 children and all leaf nodes are at the same level.

Intuition:

This is the most rigid and symmetrical type of binary tree. It's essentially a full binary tree where all the leaves are at the same depth. This structure guarantees that the number of nodes grows exponentially with the height, specifically, a perfect binary tree of height h has exactly 2h+1 − 1 nodes. A perfect binary tree is always both full and complete.

Perfect Binary Tree Example

4. Balanced Binary Tree

A Balanced Binary Tree is a binary tree in which the difference in height between the left and right subtrees of any node is at most 1.

What is a Balanced Binary Tree?

Formally, for every node in the tree:
| Height(Left Subtree) - Height(Right Subtree) | ≤ 1

This condition ensures the tree stays bushy and avoids degenerating into a list-like structure.

Why is Balance Important?

If a tree becomes unbalanced (e.g., all nodes go to one side), it can act like a linked list. This leads to operations like search and insertion taking O(N) time instead of the optimal O(log N).

Height Comparison

Tree Type Height (Worst Case) Efficiency
Balanced Tree O(log N) Fast
Unbalanced Tree O(N) Slow

Examples of Balanced Binary Trees

  • AVL Tree: Automatically balances itself with rotations after insertions/deletions.
  • Red-Black Tree: Used in libraries like Java's TreeMap or C++ STL map.

Example

Balanced Binary Tree:

        10
       /  \
      5    15
     / \     \
    2   7     20
  

All nodes have left and right subtrees with height difference ≤ 1.

Unbalanced Binary Tree:

        10
          \
           15
             \
              20
  

This tree behaves like a linked list and is not balanced.

Key Characteristics

  • Keeps operations fast: O(log N)
  • Looks like a bush, not like a stick
  • Used in self-balancing trees like AVL and Red-Black Trees
Balanced Binary Tree Example

5. Degenerate (or Pathological) Binary Tree

A Degenerate Binary Tree is a binary tree where each parent node has only one child. This results in the tree looking like a linked list.

Intuition:

This is the worst-case scenario for many binary tree operations. If you have N nodes in a degenerate tree, the height is N-1 (or N depending on definition). This means searching for an element could require traversing all N nodes, degrading performance to O(N), similar to a linear search in a linked list or array.

Left-Skewed Degenerate Binary Tree Example

Conclusion

Understanding these different types helps you recognize the structural properties of a binary tree, which is crucial for choosing the right type for a specific problem or for analyzing the performance of tree-based algorithms. Whether a tree is full, complete, perfect, balanced, or degenerate significantly impacts its efficiency characteristics.

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Types of Binary Trees

Types of Binary Trees

Full Binary Tree Complete Binary Tree Perfect Binary Tree Balanced Binary Tree Degenerate Binary Tree Types of Binary Trees Welcome back to our tutorial series on Data Structures and Algorithms! So far, we've laid the groundwork by understanding the basics of Trees and Binary Trees and then delved into some key