The Gemara (Berachot 36a) discusses the blessing for hearts of palm, the soft inner core of palm stems, and concludes that it should be “Shehakol”, and not “HaAdama”. The rationale is that palm trees are not grown specifically to harvest the inner core of the stems but rather that they are a byproduct. Nowadays, there is an industry around growing palm plants specifically to harvest the hearts, and many Poskim rule that the blessing is not “Shehakol”. Rabbi Moshe Heinemann of the Star-K suggests that, since the main “fruits” of these young palm trees are the hearts, the blessing is not even “HaAdama”, like one opinion of the Gemara, but rather “HaEtz”. Nevertheless, since heart of palm in not an actual fruit but rather a component of the stem, which happens to be edible, practically speaking the blessing is “HaAdama”. One proof for this approach is that the Ramban (Berachot 30b) explains that, although sugarcane is grown specifically for sugar, after all, the sugar that is consumed is not a fruit, but rather a product of the actual cane, and its blessingis “HaAdama”. Furthermore, the Kol Bo (end of § 204) discusses an edible flower, which although emerges from the branches of a tree like a fruit, it is considered to be a part of the structure of the tree and its blessing would aso be “HaAdama”. Yet another proof is the Mishna Berura (Sha’ar HaTziyun, O.H. 202:42) which says that any part of a tree which is not the formal fruit but rather a component of the tree, would have the blessing of “HaAdama”. Additionally, Rabbi Ilowitz, the Av Bet Din of Sao Paulo, where hearts of palm are grown, says (Birurim § 30 to Vezot HaBeracha) concurs that the proper blessing is not “HaEtz” but “HaAdama”.
The Gemara also discusses the caper bush which has several edible parts, including the actual capers, which are used as a condiment. The Shulhan Aruch (Orah Haim 202:6) rules that the blessing for the capers is “HaEtz”.
Summary: The blessing for heart of palm is “Bore Peri HaAdama”. The blessing for capers is “Bore Peri HaEtz”.