Native trees in Arizona, like palo verde and ironwood, have a tendency to form a nearly impenetrable thicket from the trunk up. That's fine out in the wild but doesn't work well in a backyard. Mesquite, which isn't a native but thrives in the desert, needs to be pruned to be kept in bounds for the average yard. Citrus trees don't require pruning, but the fruit can be kept within easy harvesting reach if the trees are cut back a bit.
Remove dead branches, leaves and debris from around the bottom of the tree.
Clip off suckers that have sprouted around the base of the tree but not from the trunk. This happens quite often with citrus trees, because in Arizona the tree is grafted onto sour orange stock, which throws up suckers. Clip any branches on the trunks up to about 7 feet for shade and specimen trees and 3 feet for citrus.
Remove dead and dying branches by trimming them off as close to the main branch as possible.
Take out branches that are brushing up against a roof, deck or tile. They could cause serious damage during the Arizona windy monsoon (rainy) season.
Examine the tree and take out branches that make the tree look lopsided. If the branch is quite long, cut it back in several sections, starting at the outermost part. Wood is heavier than it looks. Cutting the long branch could cause damage when it falls.
Cut citrus trees back to a height that you can reach by standing on a ladder. Trees that are 15 feet high are difficult to harvest. The fruit falls to the ground but isn't edible by that time. The rotting fruit attracts rats and other critters. Snakes are attracted to the rats. Arizona has rattlesnakes, even in some residential areas.
Remove branches that crisscross through the middle of the tree or rub against each other.