Arizona is the number one spot for wintering Red-naped Sapsucker in the country. But we don't have many of them in the Tucson Valley; their preferred winter habitat is oak woodland. The Atascosa Highlands and Prescott CBCs (47 last year) usually contend for the highest count each year.
But we do have some, and knowing how to find them might result in the discovery of something rarer, such as Yellow-bellied or Red-breasted Sapsucker.
There are two species of trees in town that seem to be favored by sapsuckers. One is a species of Eucalyptus, probably E. microtheca, the Coolibah. It's a lot smaller than the huge, white-barked Red River Gum (E. camaduelensis) that is such a prominent tree in the Tucson cityscape. The leaves of the one you're looking for are longer and bluer and the plant more of a weeping character. The bark of the main trunk is more scaled or flaky, rather than white and smooth. A surprisingly large number of these eucs are peppered with the holes of sapsuckers. (Click on the photos for larger images.)
Since it doesn't get so large, it's often planted around townhomes and along edges of parking lots.
The other is another Australian native, Bottle Tree (Brachychiton populneus). I have no idea what sort of sap-sucking creature utilizes this tree in Australia, as they have no woodpeckers – probably a marsupial of some sort. But it's a rare one in Tucson that doesn't have its green trunk ravaged by sapsucker. The tree looks vaguely like an evergreen cottonwood with a pudgy, green bark and odd woody fruit capsules. Check these trees thoroughly as you walk residential neighborhoods and parks.