The tree diagram (also often called a Stammbaum, or family tree) probably reveals a range of properties of language. For me, what it shows most is a sense of languages as belonging to metaphorical families, with older (dead) languages serving as parents, grandparents, or distant and all but forgotten ancestors.
If you look at a language tree that shows the ancestry (again, metaphorically speaking) of English, you’ll see that English is grouped as a West Germanic language (alongside Dutch, which is closely resembles, and German, which it more distantly resembles). This grouping can help predict with some accuracy how easy the vocabulary and grammar of another language will be to learn or even to understand with careful listening and reasoning. For an English speaker, Dutch should be a little easier to learn than German (because the grammar and vocabulary are more similar), and both Dutch and German should be easier to learn than Italian (which is also an Indo-European language but which is grouped not under “Germanic” but rather under “Italic”). In turn, all Indo-European languages should be much easier for a speaker of English to learn than, say, Chinese or Swahili.
The family tree model fails to function, though, when we look at the huge influence of languages on one another because of historical and cultural events. French had an enormous influence on English following the Norman invasion of 1066, just as American English today has an enormous influence on languages around the world.