I emailed my very favorite Stanford alumnus with the questions that Sparteye posed, and he responded thusly:

Stanford is located next to Palo Alto, which means "Tall Tree" in Spanish. There is, in fact, a tree between downtown and the Stanford Shopping Center called "El Palo Alto," but I have no idea if it is the actual eponymous tree. The university seal bears the likeness of this tree, or one similar.

Officially, the Stanford teams bear the name The Cardinal. This refers to the red color of their uniforms. In practice, it also functions as something of a shibboleth: A speaker who knows not to call the team "The Cardinals" can't be *completely* clueless. The Cardinal was adopted when "The Stanford Indian" was abrogated by the university administration in the mid-70s, IIRC. (Some likenesses of the hatchet-nosed Indian mascot still turn up at homecoming every year and even I, no real PC revisionist sympathizer, would agree that it was pretty offensive.) Why a tree, then? Legend has it that the students were actually polled and selected "The Stanford Robber Barons" to more accurately reflect the school's provenance, but couldn't get the university administration on board. Despite its lack of panache, the tree was an obvious and patently inoffensive fallback choice.

One thing that you must bear in mind is that The Tree mascot is a member of the Leland Stanford Junior University Marching Band (LSJUMB), an entirely student-run organization that enjoys only passing endorsement from the university. The tree costume changes every year, in fact, because submitting plans for a new costume is part of the audition process. Typically, the band, chafing under the obvious logistical difficulties inherent in iconifying "The Cardinal," adopted The Tree more or less in spite of the University's lack of an official stance on the issue. The story of The Dollies, the dance team that fronted the band in the absence of (more recently, in spite of) university-sponsored cheerleaders, is undoubtedly similar.

Oh, and before you punsters get ahold of this, i note the irony of his choice of the word "iconifying"