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Chapter 12: Problem 4

How many leaves does a full binary tree have if its height is (a) 3 ? (b) 7 ? (c) 12 ? (d) \(h\) ?

Short Answer

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(a) The full binary tree with height 3 has 8 leaves. (b) The full binary tree with height 7 has 128 leaves. (c) The full binary tree with height 12 has 4096 leaves. (d) For a general case, a full binary tree with height \(h\) will have \(2^{h}\) leaves.

Step by step solution

01

Understand the Question

We are asked to find the number of leaf nodes in a full binary tree given its height.
02

Use the Formula of Full Binary Tree

The number of leaves in a full binary tree with height \(h\) is given by the formula: Number of leaves = \(2^{h}\) .
03

Calculate for (a) height = 3

Using the formula, we calculate the number of leaf nodes when the height is 3: Number of leaves = \(2^{3}\) = 8.
04

Calculate for (b) height = 7

Similarly, we calculate the number of leaf nodes when the height is 7: Number of leaves = \(2^{7}\) = 128.
05

Calculate for (c) height = 12

For the height of 12: Number of leaves = \(2^{12}\) = 4096.
06

Express Number of Leaf Nodes in Terms of Height

To generalize, we can say that if a full binary tree has height \(h\), the number of leaves will be \(2^{h}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binary Tree
A binary tree is a fundamental data structure in computer science where each node has at most two children. These nodes are often referred to as the "left" and "right" children. The binary tree structure is widely used in applications such as database indexing and expression parsing.
  • The most basic type of binary tree is an unbalanced one, where the nodes have varying numbers of children from none to two.
  • Another type of binary tree is a complete binary tree, where all levels are fully filled except possibly for the last level, which is filled from left to right.
  • A full binary tree is a special kind of binary tree where every node other than the leaves has exactly two children. This means the tree is "full," and every level (except possibly the last) is completely filled.
Understanding these different kinds of binary trees is important when learning about more complex structures and navigating algorithms that require tree data manipulation.
Height of a Tree
The height of a tree is a significant concept because it influences the tree's efficiency in operations like searching and insertion. The height of a binary tree is defined as the longest path from the root node down to the farthest leaf node.
  • The height of an empty tree is -1, as it contains no nodes.
  • The height of a tree with a single node (just the root) is 0.
  • For a full binary tree with height \(h\), this translates to having multiple levels with \(h+1\) levels in total from the root to the leaves.
The height can significantly affect performance, particularly in trees that are not balanced, as unbalanced trees can degrade to similar performance metrics to that of a linked list if all nodes are on one limb.
Leaf Nodes
Leaf nodes in a binary tree are the terminal nodes of the tree. These nodes have no children and mark the end points of the tree's structure. Understanding leaf nodes is crucial because they define the output end for many tree operations.
  • In any binary tree, leaf nodes are a good indicator of the tree's base data holding capacity.
  • For a **full binary tree**, the formula for calculating the number of leaf nodes is particularly simple. Using the height \(h\), you can find the number of leaf nodes with the formula \(2^{h}\).
  • This exponential nature indicates how quickly the number of leaves increases with height, essentially doubling with every additional level.
  • Difficulty in managing leaf nodes increases with the probability of height imbalance, which can affect overall tree operations.
Leaf nodes help in limiting the functional depth of operations, and their count explains the potential breadth that tree-based searches might need to cover.

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Most popular questions from this chapter

a) At a men's singles tennis tournament, each of 25 players brings a can of tennis balls. When a match is played, one can of balls is opened and used, then kept by the loser. The winner takes the unopened can on to his next match. How many cans of tennis balls will be opened during this tournament? How many matches are played in the tournament? b) In how many matches did the tournament champion play? c) If a match is won by the first opponent to win three sets, what is the maximum number of sets that could have been played (by all entrants) during the tournament?

Let \(G=(V, E)\) be a loop-free undirected graph. Define the relation \(9 R\) on \(E\) as follows: If \(e_{1}, e_{2} \in E\), then \(e_{1} \Re e_{2}\) if \(e_{1}=e_{2}\) or if \(e_{1}\) and \(e_{2}\) are edges of a cycle \(C\) in \(G\). a) Verify that \(\mathscr{\text { is an equivalence relation on }} E\). b) Describe the partition of \(E\) induced by 9 .

The connected undirected graph \(G=(V, E)\) has 30 edges. What is the maximum value that \(|V|\) can have?

If \(G=(V, E)\) is a loop-free undirected graph, we call \(G\) color-critical if \(\chi(G-v)<\chi(G)\) for all \(v \in V\). (We examined such graphs earlier, in Exercise 17 of Section 11.6.) Prove that a color-critical graph has no articulation points.

Let \(G=(V, E)\) be a loop-free undirected graph. If \(\operatorname{deg}(v) \geq 2\) for all \(v \in V\), prove that \(G\) contains a cycle.
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