Fishing in Yellowstone National Park is regulated by the Park Service. You do not need a state license, but you do need a park fishing permit.
There are two general regulations: the Native Trout Conservation Area (green streams), which favors cutthroat trout, and the Nonnative Trout Tolerance Area (dark red/magenta), which tolerate non-native trout. There are also special regs in some areas which require you to kill non-native trout in certain streams. It is essential to read park regulations carefully.
Also, unusual for Yellowstone are the Bear Management Areas, which restrict or prohibit travel in certain areas when bears are likely to be active (and potentially dangerous). The lighter green and magenta streams indicate those within a BMA, for Native and Non-Native general reg zones, respectively.
Bright red streams are permanently closed to fishing
Click on a stream line in Google Earth to view the database attributes of the stream. Conservation Pops of Yellowstone cutthroats have a substantial amount of information.
Yellowstone Cutthroat data via StreamNet: 2007 Rangewide Assessment by the Cutthroat Conservation Group. Fishing Regulations per 2014 Yellowstone Fishing Regulations. BMA info per NPS website.