2nd Grade Tree Adding Puzzle Worksheet - Find the missing numbers in tree diagrams where each number is the sum of the two below it.

Problem Description:

The task involves solving a "Tree Adding Puzzle" where each number in the tree is the sum of the two numbers directly below it. The goal is to fill in the missing numbers in the trees so that the rule holds true for all levels.

Rules:

1. Each number in the tree is the total of the two numbers directly below it.
2. There may be multiple valid solutions for some trees.
3. The first tree has been completed as an example.

Trees to Solve:

We need to solve the remaining five trees. Let's go through each one step by step.

---

#### Tree 1 (Completed Example):
```
21
/ \
9 12
/ \ / \
2 7 5 7
```
- Verification:
- \( 9 = 2 + 7 \)
- \( 12 = 5 + 7 \)
- \( 21 = 9 + 12 \)

This tree is correctly filled.

---

#### Tree 2:
```
?
/ \
5 6
/ \ / \
3 2 4 2
```
- Step 1: Calculate the top number.
- \( 5 = 3 + 2 \)
- \( 6 = 4 + 2 \)
- Top number = \( 5 + 6 = 11 \)

- Solution:
```
11
/ \
5 6
/ \ / \
3 2 4 2
```

---

#### Tree 3:
```
?
/ \
3 8
/ \ / \
? ? 4 4
```
- Step 1: Determine the bottom numbers.
- Let the bottom numbers be \( a \) and \( b \) for the left branch, and \( 4 \) and \( 4 \) for the right branch.
- For the right branch: \( 8 = 4 + 4 \) (already satisfied).
- For the left branch: \( 3 = a + b \).

- Step 2: Choose values for \( a \) and \( b \) such that \( a + b = 3 \). Possible pairs are \( (1, 2) \) or \( (2, 1) \).

- Step 3: Calculate the top number.
- If \( a = 1 \) and \( b = 2 \):
- Top number = \( 3 + 8 = 11 \).

- Solution:
```
11
/ \
3 8
/ \ / \
1 2 4 4
```

---

#### Tree 4:
```
12
/ \
8 ?
/ \ / \
5 3 ? ?
```
- Step 1: Determine the right branch.
- Top number = \( 12 \)
- Left branch sum = \( 8 \)
- Right branch sum = \( 12 - 8 = 4 \)

- Step 2: Choose values for the bottom numbers of the right branch.
- Let the bottom numbers be \( a \) and \( b \).
- \( a + b = 4 \). Possible pairs are \( (1, 3) \), \( (2, 2) \), or \( (3, 1) \).

- Step 3: Fill in the tree.
- If \( a = 1 \) and \( b = 3 \):
```
12
/ \
8 4
/ \ / \
5 3 1 3
```

---

#### Tree 5:
```
15
/ \
? 5
/ \ / \
4 ? ? 5
```
- Step 1: Determine the left branch.
- Top number = \( 15 \)
- Right branch sum = \( 5 \)
- Left branch sum = \( 15 - 5 = 10 \)

- Step 2: Choose values for the bottom numbers of the left branch.
- Let the bottom numbers be \( 4 \) and \( b \).
- \( 4 + b = 10 \)
- \( b = 6 \)

- Step 3: Determine the right branch.
- Bottom numbers are \( a \) and \( 5 \).
- \( a + 5 = 5 \)
- \( a = 0 \)

- Solution:
```
15
/ \
10 5
/ \ / \
4 6 0 5
```

---

#### Tree 6:
```
20
/ \
6 3
/ \ / \
? ? ? ?
```
- Step 1: Determine the bottom numbers.
- Left branch: \( 6 = a + b \)
- Right branch: \( 3 = c + d \)
- Top number = \( 6 + 3 = 20 \) (This is a contradiction since \( 6 + 3 = 9 \neq 20 \)). There might be a mistake in the problem setup or additional constraints are needed.

---

Final Solutions:

1. Tree 2:
```
11
/ \
5 6
/ \ / \
3 2 4 2
```

2. Tree 3:
```
11
/ \
3 8
/ \ / \
1 2 4 4
```

3. Tree 4:
```
12
/ \
8 4
/ \ / \
5 3 1 3
```

4. Tree 5:
```
15
/ \
10 5
/ \ / \
4 6 0 5
```

5. Tree 6: Requires clarification due to inconsistency.

Boxed Final Answer:

\[
\boxed{
\begin{array}{c}
\text{Tree 2:} \\
\begin{array}{ccc}
& 11 & \\
5 & & 6 \\
3 & 2 & 4 & 2 \\
\end{array} \\
\text{Tree 3:} \\
\begin{array}{ccc}
& 11 & \\
3 & & 8 \\
1 & 2 & 4 & 4 \\
\end{array} \\
\text{Tree 4:} \\
\begin{array}{ccc}
& 12 & \\
8 & & 4 \\
5 & 3 & 1 & 3 \\
\end{array} \\
\text{Tree 5:} \\
\begin{array}{ccc}
& 15 & \\
10 & & 5 \\
4 & 6 & 0 & 5 \\
\end{array} \\
\end{array}
}
\]