* If you're thinking of the core of a fruit: This would be called the pulp or flesh.
* If you're thinking of the cambium layer: This is a thin layer of living tissue just beneath the bark that's responsible for growth. It's not exactly juicy, but it's the living, growing part of the tree.
* If you're thinking of the sapwood: This is the lighter-colored, outer layer of wood that carries water and nutrients. It can be somewhat soft and juicy depending on the tree.
To get a more accurate answer, you need to specify what kind of tree you're talking about and what part you're referring to.