Mesquite trees are a desert adapted, drought tolerant tree that can survive on less than 12 inches of rain per year. However, in a landscaped yard situation, unless the mesquite existed on your property to begin with, the tree will grow much faster and look better with supplemental irrigation. Even existing trees, although probably established enough not to necessarily require more water, will look better if you give them a drink during times of drought or the hottest part of the summer.
Methods of Delivering Water
If your mesquite is already on a drip system, much of the work is already done for you and it is just timing. If the tree is not on your irrigation system and you have one, you may want to extend the system to include the tree. Otherwise, you can water with a hose, or you can consider using passive rain water harvesting techniques.
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Drip System Irrigation
A mesquite tree on a drip irrigation system will need a schedule that is sufficient for the size of the tree. A small tree, with a canopy of 6 feet in diameter or less, will need four hours of watering with three 2-gallon-per-hour emitters. As the tree grows, you will need to add more emitters and move them farther out to match the canopy size as it expands -- about two emitters per 2 feet of horizontal growth. You also need to lengthen the watering time by two hours per 2 feet of growth. You should water your mesquite once a week during the hot months of May through September, and twice a month the rest of the year.
Watering with a Hose
If your tree is not on an irrigation system, you can add supplemental water using a hose. You want the hose to run steady, but not so hard that it runs off and away from the root zone. The hose should be placed underneath the canopy area, not at the base of the tree, since that is where the water absorbing roots are located. Let the hose run until the water soaks down at least 3 to 5 feet. Move the hose around the tree to water in at least three spots. Building up a reservoir around the tree by building a well with surrounding soil will help, too. Run the hose longer and fill up the well, letting water soak it deeply. Use the same schedule as the drip system method.
Passive Rain Water Harvesting
If you have a rain gutter and downspout on your home, and it is in an area where directing rainwater to your mesquite is feasible, by all means take advantage of the free water. All you need to do is dig a channel from the downspout and direct the water to your tree. You want the water to slow down and water the tree, but make sure it still can leave your yard so you don't end up creating a dam and a flood.