To show that the minimum height of a binary tree with $n$ vertices is $\left\lceil \log_2 (n + 1) - 1 \right\rceil$, where $\lceil x \rceil$ denotes the smallest integer greater than or equal to $x$, we need to approach this problem step by step.

1. Height of a Binary Tree

The height of a binary tree is the length of the longest path from the root to any leaf node. The tree is considered to have height 0 if it is empty, height 1 if it consists of a single node, and so on.

2. Binary Tree Properties

A binary tree has at most 2 children for each node. The goal is to find the minimum possible height of a binary tree with $n$ vertices.

• Complete Binary Tree: A binary tree is called complete if all levels are completely filled, except possibly for the last level, which is filled from left to right.

• Full Binary Tree: A full binary tree is one where every node has either 0 or 2 children (no node has only 1 child).

The most balanced (and hence minimal height) tree for $n$ vertices is a complete binary tree. In such a tree, each level is filled as much as possible.

3. Height Calculation for a Complete Binary Tree

Let’s calculate the minimum height for $n$ vertices:

• The number of nodes at each level of the binary tree follows powers of 2: the 0th level has $1$ node, the 1st level has $2$ nodes, the 2nd level has $4$ nodes, and so on.

The total number of nodes at height $h$ of a complete binary tree can be expressed as: $\text{Total nodes} = 2^0 + 2^1 + 2^2 + \dots + 2^h$ This is a geometric series, and the sum of the series gives the total number of nodes up to level $h$: $S_h = 2^{h+1} - 1$ For a complete binary tree, the number of nodes $n$ satisfies: $n \leq 2^{h+1} - 1$ Thus, to find the minimum height $h$ for a tree with $n$ nodes, we solve: $2^{h+1} - 1 \geq n$ $2^{h+1} \geq n + 1$ Taking the logarithm base 2 of both sides: $h + 1 \geq \log_2 (n + 1)$ $h \geq \log_2 (n + 1) - 1$ Since $h$ must be an integer, the minimum height of the tree is: $h = \lceil \log_2 (n + 1) - 1 \rceil$

4. Conclusion

Thus, we have shown that the minimum height of a binary tree with $n$ vertices is $\lceil \log_2 (n + 1) - 1 \rceil$.

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Would you like a deeper explanation of any step? Or do you have other related questions?

Here are 5 questions to expand on this concept:

1. How does the height of a full binary tree differ from that of a complete binary tree?
2. What is the maximum height of a binary tree with $n$ vertices?
3. How do you calculate the height of a binary tree in practice using recursion or iteration?
4. Can a binary tree with $n$ vertices have a height less than $\lceil \log_2 (n + 1) - 1 \rceil$?
5. What is the time complexity of inserting a new node in a binary tree based on its height?

Tip: For large binary trees, consider calculating heights and node distribution using logarithmic properties to optimize tree operations!