Note: Info comes from wiki article
My sprout is to yearn when hunger's its worst,
"The word "pineapple" in English was first recorded to describe the reproductive organs of conifer trees (now termed pine cones)"
When hungry, I crave something, or I guess pine for food?
And a slice of a thing you might find in a hearse.
Perhaps a cut of wood used for a pine wood casket.
My core is a dad jumbled up and inversed,
"dad" becomes "app", which is in pineapple...
And rings on your phone when you need to converse.
... and the phone portion is now an app on most smart phones, so the app would do the ringing?
Thanks to @deep thought
My leaf is an article both forward and reversed,
le and el as
"An article is a word used to modify a noun, which is a person, place, object, or idea"
Found with bananas when the bees have dispersed.
bananas without b's is "ananas"
"In the scientific binomial Ananas comosus, ananas, the original name of the fruit, comes from the Tupi word nanas, meaning "excellent fruit", as recorded by André Thevet in 1555, and comosus, "tufted", refers to the stem of the fruit. Other members of the genus Ananas are often called pine, as well, in other languages."
Interesting side note: "ananas" is actually french for pineapple. Thus, I'm thinking this was actually going for "le ananas"? But what I found suggests "l'ananas", so I dunno.
Thus, we have so far: (again, per @deep thought)
My whole's from the trees but can't fit in a purse,
Pineapples grow on trees but are too large for most purses.
Give me a squeeze and I'll cure your thirst.
Pineapple juice is delicious.