Question

The depth of a vertex in a rooted tree is defined to be the number of edges on the unique path to the root. The height of a rooted tree is the maximum of the depths of its vertices. A binary tree is complete if it is full and all its leaves have the same depth. How many vertices does a complete binary tree of height 1 have? Height \(2 ?\) Height \(d ?\) (Proof required for height \(d\).)

Short answer

For a complete binary tree, height 1 has 3 vertices, height 2 has 7 vertices, and height \(d\) has \(2^{d+1} - 1\) vertices.

Step-by-step solution

Understanding the Complete Binary Tree

A complete binary tree is a special type of binary tree where every level, except possibly the last, is completely filled, and all leaves are at the same level. It is both full and complete, so each node has either 0 or 2 children.

Analyzing Height 1

For a complete binary tree of height 1, the tree consists of a root and potentially its two children. Since height 1 implies two possible levels (the root being at level 0 and one more level), enumerate the nodes: the root (1 vertex) and its children (2 vertices), totaling 3 vertices.

Calculate Number of Vertices for Height 2

For a tree of height 2, the root has 2 children, and each of these children can have 2 more children. This adds up to 1 (root) + 2 (children of root) + 4 (children of those 2 children) = 7 vertices.

Formula Derivation for General Height d

The number of vertices for a complete binary tree of height \(d\) can be calculated using the geometric series formula for the sum of powers of 2: \(1 + 2 + 4 + \ldots + 2^d\). This is a geometric progression where the number of vertices is given by \(2^{d+1} - 1\).

Proof of the Formula for Height d

To prove that the number of vertices in a complete binary tree of height \(d\) is \(2^{d+1} - 1\), consider that for each level \(i\) from 0 to \(d\), there are \(2^i\) vertices. The sum of this geometric series is: \(\sum_{i=0}^{d} 2^i = 2^{d+1} - 1\), which is derived from the formula for the sum of a geometric series, \(S_n = a(1 - r^n)/(1 - r)\), where \(a = 1\) and \(r = 2\).

Key concepts

Complete Binary Tree

A complete binary tree is a special form of a binary tree which is both full and complete. This means that all levels, except possibly the last one, are fully filled. In simpler terms, each node at every level has two children, except for the last level, which is filled from left to right.
This structure is essential because it ensures minimal height for the number of nodes, optimizing operations such as tree traversals.

• Every level of the tree, except possibly the last, must have the maximum number of nodes.
• If there are nodes on the last level, they are aligned to the left.

In a complete binary tree, all leaves (nodes without children) are at the same depth, ensuring both balance and efficiency in storage and retrieval operations.

Tree Height

Tree height is a fundamental concept in understanding tree structures, particularly in binary trees. It is defined as the maximum number of edges in the longest path from the root to any leaf.
For a complete binary tree, the tree height corresponds to all potential levels being filled.
When considering operations or calculating properties of a tree, the height determines efficiency.

• In a binary tree, height influences the time complexity of operations such as insertion, deletion, and traversal.
• The height of a complete binary tree of height 1 would have one root and its children, leading to two levels.

Understanding the height allows for effective manipulation and utilization of binary trees in data structures.

Geometric Series

The number of vertices in a complete binary tree can be elegantly calculated using the concept of geometric series. A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the ratio.
For binary trees, the ratio is 2, as each node can have two children.
Consider the sequence for a tree's levels: 1 (root), 2 (children of root), 4 (children of children), and so on.

• This series is represented as: 1 + 2 + 4 + ... + 2^d.
• The formula for the sum of this series is a crucial tool, expressed as: \[ S_d = 2^{d+1} - 1 \]

Understanding this formula helps us predict the total number of vertices efficiently across any height of a complete binary tree.

Rooted Tree

A rooted tree is a fundamental structure in computer science and mathematics where a single node is designated as the root. This root node provides a natural hierarchy and direction for the tree, influencing operations and traversal methods.

• In a rooted tree, every node except the root has exactly one parent, establishing clear paths.
• The concept of depth is tied to the rooted tree, where depth of a node indicates its distance from the root.

The rooted tree model also plays a role in organizing data in hierarchical systems, making it a valuable abstraction for representing structured information.