Step 1: Understanding the Concept:
Homology indicates common ancestry, whereas analogy indicates similar function without common origin.
Step 2: Detailed Explanation:
1. Homology: Both the thorns of Bougainvillea and the tendrils of Cucurbita arise from the same position (axillary bud). Thus, they have a similar anatomical origin.
2. Functional Difference: Thorns are for protection, while tendrils are for climbing/support.
3. Evolutionary Process: This is called Divergent Evolution. In this process, the same structure developed along different directions due to adaptations to different needs.
Step 3: Final Answer:
They are homologous structures. Divergent evolution is the process responsible for their development.