Yes, you can grow a small persimmon tree in an outdoor plant box, but it requires careful planning and proper care. Persimmon trees, especially dwarf or compact varieties like 'Fuyu' or 'Izu,' are well-suited for container gardening. Here’s what you need to know:

1. Choose the Right Variety: Opt for dwarf or semi-dwarf persimmon trees, as they adapt better to confined spaces.

2. Select a Large Plant Box: Ensure the container is at least 20-24 inches deep and wide to accommodate root growth.

3. Use Well-Draining Soil: Persimmons thrive in loamy, slightly acidic soil with good drainage to prevent root rot.

4. Provide Full Sunlight: Place the plant box in a spot that receives 6-8 hours of direct sunlight daily.

5. Water and Fertilize Regularly: Keep the soil consistently moist but not waterlogged, and use a balanced fertilizer during the growing season.

6. Winter Protection: In colder climates, insulate the plant box or move it to a sheltered area to protect the tree from frost.

With proper care, your persimmon tree can thrive in an outdoor plant box, yielding delicious fruit in a few years. Happy gardening!