Example of a probability tree diagram for two fireworks, showing outcomes and probabilities.

Problem Analysis:

The problem involves calculating the probability that both of John's fireworks will light up. We are given the following information:

1. The probability that a particular brand of firework lights is \( \frac{7}{8} \).
2. The probability that a particular brand of firework fails to light is \( \frac{1}{8} \).
3. John buys two fireworks, and we need to use a tree diagram to find the probability that both fireworks light up.

Step-by-Step Solution:

#### Step 1: Understand the Tree Diagram
The tree diagram represents the outcomes for two fireworks:
- The first branch represents the outcome of the first firework (Lights or Fails).
- The second set of branches represents the outcome of the second firework (Lights or Fails), depending on the outcome of the first firework.

#### Step 2: Fill in the Probabilities
We know:
- Probability that the first firework lights: \( \frac{7}{8} \)
- Probability that the first firework fails: \( \frac{1}{8} \)
- Probability that the second firework lights: \( \frac{7}{8} \) (independent of the first firework)
- Probability that the second firework fails: \( \frac{1}{8} \) (independent of the first firework)

#### Step 3: Identify the Desired Outcome
We are interested in the scenario where both fireworks light up. This corresponds to the path in the tree diagram where:
- The first firework lights (probability = \( \frac{7}{8} \))
- The second firework lights (probability = \( \frac{7}{8} \))

#### Step 4: Calculate the Probability of Both Fireworks Lighting Up
Since the events are independent, the probability of both fireworks lighting up is the product of their individual probabilities:
\[
P(\text{Both Light}) = P(\text{First Lights}) \times P(\text{Second Lights})
\]
Substitute the given probabilities:
\[
P(\text{Both Light}) = \frac{7}{8} \times \frac{7}{8}
\]

#### Step 5: Perform the Multiplication
\[
P(\text{Both Light}) = \frac{7}{8} \times \frac{7}{8} = \frac{7 \times 7}{8 \times 8} = \frac{49}{64}
\]

#### Step 6: Verify the Tree Diagram
The tree diagram already shows this calculation:
- The path "First Lights" and "Second Lights" has a combined probability of \( \frac{7}{8} \times \frac{7}{8} = \frac{49}{64} \).

Final Answer:

The probability that both of John's fireworks light up is:
\[
\boxed{\frac{49}{64}}
\]