REVISION/ADDITION: Furthermore, as mentioned by Suzuki-san, the katakana here allows people a simpler way to read the name of the tree, which is most likely placed in a public park frequented by children. Kanji might actually prevent less knowledgeable individuals from reading the sign, so the decision to use katakana here is also inspired by the context in which the tree is placed, though hiragana would also suffice for this purpose. When considered in the context of the other reasons to use katakana, though, I can see why the sign turned out the way it did.
REVISION/ADDITION: As to the choice to use katakana for this girl's name (ナミ）, specifically, given that the name "Nami" is a common one in Japan, is to specifically enhance the reader's feeling that the story's setting and characters are not Japanese in origin (later in the manga, when characters specifically intended to seem Japanese appear, their names are spelled in Kanji and Hiragana). Thus, in this case, katakana serves to differentiate Nami from a traditional Japanese character, despite naming her after a Japanese word.
2. The reason that these types of effects exist is because katakana represent a fundamentally different alphabet from hiragana, which are often the alternative to their use While the two sound identical in almost every respect, because katakana have a different visual representation and a different role in Japanese language, they can be used to create a different effect than hiragana even if there is no difference in the audible nature of the words they are used to spell. Japanese writers are clearly aware of this, and now intentionally use katakana because of its unique qualities, just as they use hiragana and kanji for other things. In summary, "katakana effects" exist because people feel different things about words when they are spelled in katakana than if they are spelled any other way.
REVISION/ADDITION: Since the question was raised in my comments, I will clarify that I feel that the exact scope of these effects is something impossible to quantify; as my katakana examples and the examples of others show, the uses of katakana and the goals those uses fulfill are nearly limitless. However, I feel that the important consideration to make here is that the reason that a reader and writer of Japanese feels two different things when they consider a word spelled in katakana - as compared to hiragana and kanji - is because the significance of katakana is larger than the sounds the characters represent. Katakana occupy a fluid, dynamic space within Japanese linguistic culture, and the work of actually defining all their unique effects (or even laying out basic rules and patterns for them) is an extremely difficult process.
3. The textbooks we read were actually very similar in their explanations of katakana. They almost all argued that katakana's primary purpose was to spell loanwords and foreign names, rather than Japanese words. Despite the fact that this explanation of katakana is incomplete, I can understand why they chose to explain them that way. Katakana are actually an extremely nuanced facet of Japanese language, and their meaning is a rather dynamic concept. Sometimes katakana are used in ways that intentionally break with the common methods of their use, and those breaks would be difficult to capture in a textbook. However, I would suggest that textbooks also include a passage that provides examples of alternative uses of katakana and cautions individuals learning Japanese that there are no set-in-stone rules about them, in order to broaden their perspective on Japanese language and to prevent them from becoming confused when they are faced with uses of katakana like the above ones.
REVISION/ADDITION: It is also worth noting, as one of my readers mentioned, that such "alternative katakana education" should be contextualized within the level of Japanese language knowledge it is delivered in. While katakana does have many interesting uses, it is undeniable that its "primary" purpose is still for the use of loanwords in everyday speech. Thus, particularly in extremely basic Japanese education, it may not be as productive to pedagogically engage with the more complex and multifaceted uses of katakana until the student has achieved a greater understanding of Japanese language and culture. However, I still maintain that even beginner Japanese students could benefit from learning more in depth applications of katakana to some extent.
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