At the nursery, the trees they sell are labeled as 'fruiting' , 'flowering' , 'shading' 

What are the 'flowering' ones biological purpose ?  Are these somehow manipulated trees; or their flowers have no stamen, stigma, no reproduction purpose ?  Those do not bear fruit.  (Like the ornamental cherry blossoms) How does it work ?

Human driven artificial selection AFAIK

2 hours ago, Externet said:

What are the 'flowering' ones biological purpose ? 

Reproduction of course. Some trees are male, some are female and some are both. Ornamental trees are often manipulated to produce showy flowers and/or minimal fruit. I've never heard of a flowering tree that produced no fruit at all.

Most likely what you are seeing is not based around anatomy but around its purpose. Fruiting labels on trees most commonly mean sizable edible fruits for humans. When a label says flowering, it usually just means a tree with showy flowers, like magnolias or tulip poplar. When a tree says shading it often means a tree that is easily limbed up, with a sprawling form and taller.  If you see one that says privacy, it normally means grows quickly and is compact. Often they will grow them far to close, and after a few years remove every other one. 
 

The labels are not usually about anatomy unless it’s male or female related like with holly trees. 

Flowering trees and fruiting trees play an essential role in botany. Flowering trees showcase colorful blooms that attract pollinators, while fruiting trees produce seeds enclosed in fruits, supporting plant reproduction and feeding wildlife. Both types are vital for ecosystems, biodiversity, and human agriculture.