I’m made of two parts, you see
Where the first part is a tree
Whose trunk can be used for furniture
Its leaves may be used to brew tea
The first of its two parts, pine, is a tree whose wood is frequently used in furniture making. "Pine needles can be simmered into a tea that’s rich in vitamin C, and also happens to be pretty tasty."
The other is from another tree
Whose growth is not fast-paced
The second part, apple, is a type of tree that generally has a medium growth rate - or, more to the point, is not considered to be a fast growth rate ("The designation slow means the plant grows 12” or less per year; medium refers to 13 to 24” of growth per year; and fast to 25” or greater", and most apple trees are considered "medium" growth rate on this scale—though there are indeed some that are classed as "fast").
Together, I relate not to a tree
A misnomer, yet I too have a taste
Pineapple plants are shrubs, not trees.
It is misnamed in that, originally, a 'pine apple' was what we now call a pine cone - which would indeed be related to a tree. But ... Because English ... we (quite anomalously) now call the fruit and its plant a "pineapple" because of the fruit’s general resemblance to a pine cone, whereas most other languages use "ananas" or something related. See also Why is “pineapple” in English but “ananas” in all other languages?.
Pineapples, of course, have their own taste, one I quite enjoy.