You know western Wyoming and dumb luck are both on your side when:

• Your daughter spies three mule deer in a Yellowstone meadow. Then a moose mid-river. Then bison, fox and marmot, trumpeter swans, a wayward seagull and a grizzly family — mama bear and two cubs, romping across the high slopes, safely distant but still riveting.

• You hear the word "rodeo" used as a verb. Then you attend one in Cody, about 50 miles east of Yellowstone, and see not only bucking broncs, bull-riding, barrel-racing and calf-roping but also a stunt rider who circles the ring while standing astride two galloping horses.

• You look up from lunch at Buffalo Bill's old hotel and find that Miss Rodeo Wyoming is seated at the counter, right between Miss Cody Stampede and Miss Rodeo America, all chowing down in their spangled blouses and sashes.

• Old Faithful, which generally rests for 90 minutes between eruptions, starts spouting the moment you step up.

• The Old Faithful Inn, whose dinner tables are often booked months in advance, has space for you the moment you step up. (It helps to step up at 10 minutes before 5 p.m.)

• On a foray into Yellowstone's Mammoth Hot Springs area, you discover an appalling array of tasteless lawn ornaments at the home reserved for the Yellowstone concessionaire's top executive. Then they move, and you realize the elk are real. All 10 of them. They can't resist the grass and shade, the camera-happy tourists can't resist the elk and the rangers are forever struggling to keep the beasts with antlers separated from the beasts without.

OK, so by now, you've realized this isn't a multiple-choice test. It's more a reminder: Even when fully besieged by the summering masses, Yellowstone National Park remains a wildlife parade, a geothermal freak show, an essential rite of North American tourism, a lot of fun. And a side trip to Cody can fit about as nicely as cornbread alongside a slab of ribs.

In early July, my family and I spent five days in tiny Cody and massive Yellowstone, whose 3,472 square miles cover much of northwestern Wyoming, spilling over into southern Montana and eastern Idaho. The park — which became the first national park in the world when it opened in 1872 — had nearly 3.3 million visitors in 2009.

That was a record, but with so many Americans reconsidering foreign travel, packing up cars and heading for the parks, it might soon be broken. Yellowstone spokesman Al Nash says June was the busiest ever at the park: 694,841 visitors in 30 days. For at least a few nights of our visit, every one of the more than 2,000 hotel rooms and cabins in the park was booked.

That means a lot of traffic along the park's 142-mile, figure-8-shaped Grand Loop Road, especially with so many animals afoot this summer, nibbling the generous growth after a wet spring. On Aug. 25, the park will add one more enticement by replacing a batch of temporary buildings with a new $26-million Old Faithful Visitor Education Center, replete with green construction elements and hands-on exhibits to show how and why the Earth here spits hot water skyward.

If you enter the park from the south, the drama starts right away, with the steep slopes down to the Lewis River, the racing water and the ridgelines crowded with charred trees remaining from the fires of 1988. An estimated 36% of the park burned during that dry summer, and legions of skeletal lodgepole pines endure, dead sticks standing. But two decades of regrowth have brought another generation along, shorter and greener, putting visitors in the middle of an epic reminder that death and rebirth are natural neighbors.

We knew to expect big crowds and occasional traffic jams (some caused by roadwork, most caused by gawking drivers and meandering park mammals) so we didn't mind them. Like generations of families before us, my wife, Mary Frances, and our 6-year-old, Grace, opened a notebook to count sightings of critters and cars. Soon they had 40 states. As I steered the rental car down the aisles of a packed lot near Old Faithful, Mary Frances sounded just a little regretful when I finally found a spot.

"This parking lot," she says, "is a gold mine of license plates!"

Fearing the "Mona Lisa" effect ("But it's so small!"), we didn't expect much from Old Faithful. But when the geyser immediately leapt into action, about 100 feet high and surrounded by perhaps 1,000 expectant visitors, there was no time for letdown. We took in the spectacle and sulfurous scent for perhaps five minutes, then marched right along to the Upper Geyser Basin, following boardwalks past scores of geothermal features, from eerily colored pools to mini-mountains spouting towers of spray.

"It smells like Milo," said Grace. (Milo is a 12-year-old golden retriever with occasional gastric issues.)

Until you get here, it may be impossible to appreciate all the ways that water rises, falls, rushes, rests, is cooked, is chilled, is channeled and is flung in Yellowstone. There are more than 300 geysers and almost as many waterfalls, including the roaring wonders known as the Upper Falls and Lower Falls of the Grand Canyon of the Yellowstone.

Watching Lower Falls, Tim Cahill writes in his book on Yellowstone, "Lost in My Own Backyard," is "like watching a fire: the same thing keeps happening and happening, but just a little differently every time, so that it holds the eye and empties the mind."

So we risked vertigo and took the half-mile switchback trail down to the Brink of Lower Falls. Spellbinding. Then we took the easy strolls to Inspiration Point and Artist Point — two more postcard views. The steeply pitched canyon may be only about 20 miles long, but to paraphrase Richard Nixon's assessment of that wall in China, it is grand indeed.