In Norse Paganism, along with most other forms of Paganism, trees are very important both in the mythology and as symbols. The most notable example of this is Yggdrasil, the huge ash tree that connects the Nine Worlds. Midgard is the world we call the "material world", Asgard is the home of the Aesir, the deities associated with war, Helheim the underworld, Nifelheim the world of fog and darkness, and so on.
Another tree related myth is the one of Askr and Embla, who were said to be the first two humans. Askr means "Ash" and Embla means "Elm". In a ritual these two deities were likely symbolized by their respective trees. There is also a theory that Embla in fact means "vine", but "Elm" seems more likely. This myth is very similar to the Christian one of "Adam and Eve", and the Christians probably adopted and inverted this legend from the Indo-Europeans, who were traveling through the area at the time.
Also, I think trees have a significance because of their age and importance to us. We cannot survive without them, thus they should be held in high regard. The fact that some are centuries old makes them "living gateways to the past" so to speak, giving them even more value.