Final answer:
Mother Shipton and the Duchess likely treat Piney Woods with affection due to her youth and innocence or a desire to offer her family-like support. Their own lives' contrasts to Piney's potential and hope could motivate their actions.
Step-by-step explanation:
The question seems to involve the analysis of characters and their motivations within a narrative, likely from a piece of literature. A key reason for Mother Shipton and the Duchess's affectionate treatment of Piney Woods could be their recognition of her youth, innocence, and hope which stand in contrast to their own lives that have perhaps been marred by more disreputable professions. They may see in her the opportunity to nurture or support someone who represents what they can no longer claim for themselves. On the other hand, it's also possible that they treat her as a beloved child because they want to provide her with a sense of family and protection, especially if Piney lacks a family of her own. The provided texts suggest an emphasis on characters navigating social and personal challenges, often with a focus on women's roles and perceptions.