The Stanford Tree is not the mascot of Stanford University. That's the cardinal (the color, not the bird, so it's the Stanford Cardinal, not the Stanford Cardinals). However, it is the Tree that rallies students and marches with the band. The Tree is known for it's general drunkenness/rowdiness at sporting events. In fact, when I was a freshman, my first night at Stanford, the Tree had to be carried off by a group of people yelling that the tree was "high on life" (high enough that it was unable to walk on it's own). Needless to say, like many other Stanford students, the Tree is one of my favorite things about the university.
A funny side note: to be the tree, you have to do go through the process of running for tree. This consists of doing increasingly stupid/crazy stunts. A panel, consisting of the school marching band, former trees, and The Dollies (Stanford's cheerleaders) elect the new tree from the group of candidates. The former trees pull the have the most say in this process. Anyway, a friend of mine freshman year (who was a drawmate sophomore year) ran for tree, and did some pretty crazy shit. He put body wax strips all over himself and let people pull them off (which was still exremely painful even after quite a bit of alcohol). He sat in one of the main fountains, wearing nothing but a strategically placed sock, filled the fountain with goldfish, and then using a shot glass, started shooting goldfish. He had his mom spank him in public (the candidacy for tree is held during parents weekend, so as to discourage everyone but the most hardcore). For his finale, he did something involving putting a live snake down his pants. Unfortuntately, he lost to some junior. Mostly her stunts weren't that great, but for her finale, she did have someone shoot her (with a real gun, but she was wearing a bullet proof vest). That's pretty fucking hardcore. Rumor has it she also covered herself in nothing but saran wrap, brought herself to the group of former trees, and said "do with me what you will." Not sure how much validity there is to that one.