Welcome to the tree planting problem! This is a classic elementary math problem that studies the relationship between the number of trees and the number of intervals when planting trees at regular intervals on a line segment or closed figure. Let's explore this fundamental concept together.
The first case is planting at both ends. This is the most common type of tree planting problem. When we plant trees at both the starting point and ending point of a line segment, we discover an important pattern: the number of trees equals the number of intervals plus one. In this example, we have 6 trees and 5 intervals, demonstrating this fundamental relationship.
The second case is planting at only one end. In this scenario, we plant trees at either the starting point or ending point, but not both. This creates a different relationship: the number of trees equals the number of intervals. Here we see 5 trees and 5 intervals, showing this one-to-one correspondence. This pattern is useful for problems like counting steps or calculating positions.
The third case is planting at neither end. Here we don't plant trees at either the starting point or ending point, only in the middle section. This creates another distinct pattern: the number of trees equals the number of intervals minus one. In this example, we have 4 trees and 5 intervals, demonstrating this relationship where trees are fewer than intervals.
The fourth case is planting on a closed figure, such as a circle or square. In this scenario, the number of trees equals the number of intervals, creating a perfect one-to-one relationship. Here we see 8 trees arranged in a circle with 8 intervals between them. Understanding these four different cases - planting at both ends, one end, neither end, and on closed figures - is the key to solving all tree planting problems successfully.