If the tree in the picture above was the word for ‚three‘, the thick branches would be Hittite teri, Latin trēs, Germanic *þrejez and so on. All of them stem from the PIE root *trei̯es. The Germanic branch splits up into Gothic þreis, Swedish tre and three (amongst others), and further into dialects.
We like to speak of abstract stuff in terms of visible things. For words, one metaphor (Gr. μετα-φέρω meta-phérō ‚to carry across‘) is trees, because they grow and branch out like words or languages. In the 19th century, the tree model was a milestone for historical linguistics. It transferred the scientific concept of evolution onto languages, inspiring linguistics, but also carrying the notion that languages are living creatures.
Modern historical linguistics considers language to be a part of human culture, a tool we form collectively, not an organism. Besides, the aim of historical linguists is not to judge wrong and right language (prescriptive), but to look at languages growing (language change) in order to find patterns (descriptive).
tl;dr: Languages are not trees, so neither are words. Both have roots and stems and branch out, but language depends on humans.