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Properties of Binary Tree - GeeksforGeeks

Last Updated : 23 Jul, 2025

This post explores the fundamental properties of a binary tree, covering its structure, characteristics, and key relationships between nodes, edges, height, and levels

Note: Height of root node is considered as 0.

A binary tree can have at most 2^l nodes at level l.

• Level Definition: The number of edges in the path from the root to a node. The root is at level 0.
• Proof by Induction:

Base case: For root (l = 0), nodes = 2⁰ = 1.

Inductive step: If level l has 2^l nodes, then the next level has at most twice as many:

2×2^l = 2^l+1

A binary tree of height h can have at most 2^h+1 - 1 nodes.

• Height Definition: The longest path from the root to a leaf node. Please note that a tree with only one root node is considered to have height 0 and an empty tree (or root is NULL) is considered to have height "-1"

Formula Derivation: A tree has the maximum nodes when all levels are completely filled. Summing nodes at each level:

1 + 2 + 4 +...+ 2^h = 2^h+1 - 1

• Alternate Height Convention: Some books consider a tree with only one root node is considered to have height 1 and an empty tree (or root is NULL) is considered to have height 0. making the formula 2^h - 1.

The minimum possible height for N nodes is ⌊log⁡₂N⌋.

Explanation: A binary tree with height h can have at most 2^h+1 - 1 nodes.

Rearranging:

N ≤ 2^h+1 − 1
2^h+1 ≥ N+1
h ≥ log_2​(N+1) - 1 (Taking log₂ both sides)
h ≥ ⌊log₂​N⌋

This means a binary tree with N nodes must have at least ⌊log⁡₂N⌋ levels.

A binary tree with L leaves must have at least ⌊log⁡₂L⌋ levels.

Why? A tree has the maximum number of leaves when all levels are fully filled.

From Property 1:
L ≤ 2^l ( l is the level where leaves appear)

Solving for l:
l_min = ⌊log⁡₂L⌋

This gives the minimum levels needed to accommodate L leaves.

In a full binary tree (where every node has either 0 or 2 children), the number of leaf nodes (L) is always one more than the internal nodes (T) with two children:

L=T+1

Proof:

A full binary tree has a total of 2^h+1 - 1 nodes.

Leaves are at the last level: L = 2^h.

Internal nodes: T =2^h (2−1) − 1= 2^h - 1.

Simplifies to L=T+1

In any non-empty binary tree with n nodes, the total number of edges is n - 1.

Every node (except the root) has exactly one parent, and each parent-child connection represents an edge.

Since there are n nodes, there must be n - 1 edges.

• Each node has at most two children.
  • 0 children → Leaf Node
  • 1 child → Unary Node
  • 2 children → Binary Node

• Full Binary Tree → Every non-leaf node has exactly two children.
• Complete Binary Tree → All levels are fully filled except possibly the last, which is filled from left to right.
• Perfect Binary Tree → Every level is completely filled, and all leaves are at the same depth.
• Balanced Binary Tree → The left and right subtrees differ in height by at most 1.

Tree traversal is categorized into Depth-First Search (DFS) and Breadth-First Search (BFS):

• DFS Traversals: Explore one branch fully before backtracking.
  • In-Order (LNR): Left → Node → Right (retrieves BST elements in sorted order).
  • Pre-Order (NLR): Node → Left → Right (used for tree reconstruction).
  • Post-Order (LRN): Left → Right → Node (helps in deleting or evaluating expressions).
• BFS Traversals: Visit nodes level by level.
  • Level-Order: Processes nodes from top to bottom (used in shortest path algorithms).
  • Zig-Zag Traversal: Alternates left-to-right and right-to-left at each level (used in hierarchical structures).

See Handshaking Lemma and Tree for proof
Different types of Binary Trees and their properties
Introduction to Binary Tree in set 1

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